Title: AP Statistics — Active Recall
Tags: diff-2 unit-5 unit-3 unit-1 unit-4 unit-2 diff-1 diff-3 lo-4.10 lo-5.3 lo-5.4 lsrl correlation-r lo-5.2 matched-pairs normal-distribution conclusion definition lo-5.1 paired-vs-two-sample scatterplots binomial errors-power lo-1.7 lo-2.7 lo-3.10 lo-4.3 summary-statistics influential-points lo-1.10

Front: What is the core idea that statistics studies, and why does it exist as a discipline?
Back: Statistics studies variation — the fact that measurements differ from individual to individual. It gives tools to describe that variation and to draw reliable conclusions from data despite it.
Extra: Why students miss this: Statistics is not just arithmetic on numbers; it is reasoning about patterns and uncertainty in data.

Front: A teacher measures the height of every student in one class and finds the values are all different. Why is this variation the reason statistical tools are needed?
Back: Because no single value describes the whole group, you need summaries (center, spread) and graphs to capture the pattern, and you must account for the spread before making any claim about students beyond that class.
Extra: Why students miss this: Variation is not measurement error to be ignored; it is the real signal statistics is built to handle.

Front: Define a categorical variable and a quantitative variable.
Back: A categorical variable records which group an individual falls into (a label or category). A quantitative variable records a numerical amount that you can meaningfully do arithmetic on, like add or average.
Extra: Why students miss this: Numbers alone do not make a variable quantitative — ZIP codes and jersey numbers are categorical because averaging them is meaningless.

Front: A survey records each person's ZIP code, number of pets, and favorite season. Classify each variable as categorical or quantitative.
Back: ZIP code is categorical (it labels a location even though it looks numeric). Number of pets is quantitative (a count you can average). Favorite season is categorical.
Extra: Why students miss this: A variable stored as digits, like ZIP code, is still categorical if arithmetic on it is meaningless.

Q: A [...] quantitative variable can take only separate, countable values, while a [...] one can take any value in an interval.
A: discrete; continuous
Extra: Discrete = countable, separate values (like number of siblings). Continuous = any value within a range (like time or weight).
Why students miss this: Continuous does not mean 'has decimals' in the data table; it means the underlying quantity can in principle take any value in an interval.

Front: What does a relative frequency table show, and how is each entry computed?
Back: It shows the proportion (or percent) of individuals in each category. Each entry is the category's count divided by the total number of individuals.
Extra: Why students miss this: Relative frequencies are fractions of the total, so they should sum to 1 (or 100%), not match the raw counts.

Front: Out of 240 surveyed commuters, 84 bike to work. What is the relative frequency of biking, as a percent?
Back: 84 ÷ 240 = 0.35, so 35% of commuters bike to work.
Extra: Why students miss this: Do not report the raw count of 84 as the relative frequency; you must divide by the total first.

Front: Two schools report how many students chose each lunch option. One school has 300 students and the other has 900. Why should you compare relative frequencies rather than raw counts?
Back: Because the schools have very different totals, raw counts are misleading — the larger school will tend to have bigger counts everywhere. Relative frequencies put both groups on the same per-student scale so the comparison is fair.
Extra: Why students miss this: A bigger raw count does not mean a category is more popular if the groups have different sizes.

Front: What kind of variable is a bar chart used for, and what does the height of each bar represent?
Back: A bar chart displays a categorical variable. Each bar's height shows the frequency or relative frequency of that category.
Extra: Why students miss this: A bar chart is for categories, not for showing the shape of a quantitative distribution — that is a histogram.

Front: Why are the bars in a bar chart drawn with gaps between them, unlike the touching bars of a histogram?
Back: The gaps signal that the categories are distinct labels with no numerical order or spacing between them. A histogram's bars touch because they cover an unbroken numerical scale.
Extra: Why students miss this: Touching versus gapped bars is not a style choice; it marks the difference between quantitative and categorical data.

Front: A researcher wants to display the proportion of survey respondents in each of four blood-type groups. Which graph type fits, and why?
Back: A bar chart (or pie chart) fits, because blood type is categorical. Bars show each group's share clearly, and a pie chart works if you want to emphasize parts of a whole.
Extra: Why students miss this: Do not reach for a histogram or boxplot here — those are for quantitative variables.

Front: What does a histogram display, and what does each bar represent?
Back: A histogram displays the distribution of a quantitative variable. The data are split into equal-width intervals (bins), and each bar's height is the count or proportion of values falling in that bin.
Extra: Why students miss this: Each bar covers a range of values, not a single value, so a histogram is not the same as a bar chart.

Front: You have the exact exam scores of 12 students and want a graph that keeps every individual value visible. Which display is a good choice, and why?
Back: A dotplot (or stemplot) works well, because with a small dataset it shows each value individually while still revealing the overall shape, center, and spread.
Extra: Why students miss this: A histogram would bin the values and hide the individual scores, which is unnecessary for such a small set.

Front: In a histogram of reaction times, the bar covering 0.20–0.30 seconds has height 14. What does that tell you?
Back: Fourteen of the measured reaction times fell in the interval from 0.20 up to 0.30 seconds.
Extra: Why students miss this: The height does not give the value of any single observation, only how many landed in that bin.

Front: Why can the same dataset look very different depending on the chosen histogram bin width?
Back: Wide bins lump many values together and smooth over detail, possibly hiding gaps or extra peaks; narrow bins show fine detail but can look choppy and exaggerate random bumps. Bin width is a choice that shapes the visual impression.
Extra: Why students miss this: There is no single 'correct' histogram; bin width is a judgment call, so describe shape cautiously.

The acronym SOCS reminds you to describe a distribution's Shape, Outliers, Center, and Spread.
Extra: Shape, Outliers, Center, Spread — the four features to address when describing any quantitative distribution.
Why students miss this: A complete description names all four features, not just the center.

Front: What does it mean for a distribution to be skewed right versus skewed left?
Back: Skewed right has a long tail stretching toward larger values (the right). Skewed left has a long tail stretching toward smaller values (the left). The skew is named for the direction of the tail.
Extra: Why students miss this: Skew direction follows the thin tail, not the bulk of the data or the peak.

Front: For a set of household incomes, the mean is much larger than the median. What does this suggest about the shape of the distribution?
Back: It suggests the distribution is skewed right: a few very high incomes pull the mean upward above the median, creating a long right tail.
Extra: Why students miss this: A mean above the median signals right skew, not left skew.

Front: When you describe the center and spread of a distribution on the AP exam, why must you state them in the context of the variable?
Back: Because a bare number like '72' is meaningless without units and subject — saying 'the median test score was 72 points' communicates what was measured. Context turns a statistic into a real conclusion.
Extra: Why students miss this: Reporting statistics without context (units and what was measured) loses credit and meaning.

Front: A dotplot of shoe sizes shows two clear separate peaks. What shape term describes this, and what might it indicate?
Back: It is bimodal. Two peaks often indicate two distinct subgroups mixed together — here, perhaps separate clusters for two different groups of people.
Extra: Why students miss this: Two peaks do not mean the data are 'wrong'; they often reveal hidden subgroups worth investigating.

Front: How is the median of a dataset found?
Back: Order the values from smallest to largest; the median is the middle value. If there is an even number of values, it is the average of the two middle values.
Extra: Why students miss this: You cannot read the median off unsorted data — ordering the values first is essential.

Front: Find the median of these reaction times (seconds): 0.9, 0.4, 0.7, 0.5, 1.2, 0.6.
Back: Sorted: 0.4, 0.5, 0.6, 0.7, 0.9, 1.2. With six values, average the two middle ones (0.6 and 0.7): (0.6 + 0.7) ÷ 2 = 0.65 seconds.
Extra: Why students miss this: Do not pick a single value as the median of an even-sized set; average the middle pair.

Front: Why is the median called resistant while the mean (x̄) is not?
Back: The median depends only on the middle position, so extreme values barely move it. The mean adds in every value, so a single very large or small observation can pull it noticeably. The median resists outliers; the mean does not.
Extra: Why students miss this: An outlier shifts the mean much more than the median, so a large gap between them flags skew or outliers.

Front: Define the interquartile range (IQR) and what it measures.
Back: IQR = Q3 − Q1, the range of the middle 50% of the data. It measures spread and, because it ignores the extremes, it is resistant to outliers.
Extra: Why students miss this: IQR is not the full range (max − min); it spans only the central 50% of values.

Front: A dataset has Q1 = 18, median = 25, Q3 = 34, minimum = 11, maximum = 52. Compute the range and the IQR.
Back: Range = max − min = 52 − 11 = 41. IQR = Q3 − Q1 = 34 − 18 = 16.
Extra: Why students miss this: Do not confuse range (uses min and max) with IQR (uses Q1 and Q3).

Front: In plain words, what does the standard deviation of a dataset tell you?
Back: It measures the typical distance of the values from the mean. A larger standard deviation means the data are more spread out; a value near 0 means the data cluster tightly around the mean.
Extra: Why students miss this: Standard deviation measures spread, not center, and it is never negative.

Front: Two classes have the same mean test score, but Class A has a standard deviation of 4 points and Class B has 12 points. What does this say about the two classes?
Back: Scores in Class A are tightly bunched near the mean, so students performed similarly. Class B's scores are much more spread out, so performance varied widely even though the averages match.
Extra: Why students miss this: Equal means do not mean equal distributions — spread can differ greatly.

Front: What five numbers make up the five-number summary, and which graph displays them?
Back: Minimum, Q1, median, Q3, and maximum. A boxplot displays them — the box spans Q1 to Q3 with the median marked inside, and whiskers reach to the smallest and largest non-outlier values.
Extra: Why students miss this: The five-number summary does not include the mean or standard deviation.

Q: By the [...] rule, a value is an outlier if it falls below Q1 − [...] or above Q3 + 1.5 × IQR.
A: 1.5 × IQR; 1.5 × IQR
Extra: A value is flagged as an outlier when it lies more than 1.5 times the IQR below Q1 or above Q3.
Why students miss this: The fences use 1.5 × IQR added past Q1 and Q3, not past the median or the mean.

Front: A dataset has Q1 = 20 and Q3 = 32. Use the 1.5 × IQR rule to decide whether a value of 55 is an outlier.
Back: IQR = 32 − 20 = 12, so 1.5 × IQR = 18. The upper fence is Q3 + 18 = 32 + 18 = 50. Since 55 > 50, it is an outlier.
Extra: Why students miss this: You must compute the fence with 1.5 × IQR; a value can look large yet still fall inside the fence.

Front: Why can a boxplot hide important features of a distribution that a histogram or dotplot would reveal?
Back: A boxplot only shows the five-number summary, so it cannot reveal gaps, clusters, or whether a distribution is unimodal or bimodal. Two very different shapes can produce nearly identical boxplots.
Extra: Why students miss this: A boxplot does not show modality or local peaks; do not claim shape detail from a boxplot alone.

Front: When the AP exam asks you to compare two distributions, what must your answer explicitly do?
Back: Use comparative language (greater than, less than, about the same) to directly contrast the two groups on shape, center, and spread, in context — not just describe each distribution separately.
Extra: Why students miss this: Describing each distribution on its own is not comparing; you must explicitly relate one to the other.

Front: Two side-by-side boxplots of commute times show Town A with median 25 min and IQR 10 min, and Town B with median 25 min and IQR 22 min. Write a comparison.
Back: The typical commute is about the same in both towns (medians both 25 minutes), but commute times are far more variable in Town B (IQR 22 minutes) than in Town A (IQR 10 minutes).
Extra: Why students miss this: Equal medians do not make the distributions equivalent; the spreads differ and must be compared.

Front: Why is it usually better to compare two groups using medians and IQRs when at least one distribution is strongly skewed?
Back: Skew and outliers distort the mean and standard deviation, so a comparison built on them can mislead. The median and IQR are resistant, giving a fairer picture of typical value and spread for skewed data.
Extra: Why students miss this: Means are not the default for every comparison; for skewed data, resistant measures are more honest.

Front: State the empirical (68–95–99.7) rule for a roughly normal distribution.
Back: About 68% of values fall within 1 standard deviation of the mean, about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations.
Extra: Why students miss this: The empirical rule applies only to distributions that are approximately normal (bell-shaped), not to every dataset.

Front: Battery lives are approximately normal with mean 40 hours and standard deviation 5 hours. About what percent of batteries last between 30 and 50 hours?
Back: 30 and 50 hours are 2 standard deviations below and above the mean (40 ± 2×5). By the empirical rule, about 95% of batteries last between 30 and 50 hours.
Extra: Why students miss this: Do not apply 68% here — 30 to 50 spans two standard deviations, not one.

Front: What does a z-score measure, and how is it computed?
Back: A z-score tells how many standard deviations a value lies above or below the mean. Compute it as (value − mean) ÷ standard deviation. A negative z-score means the value is below the mean.
Extra: Why students miss this: A z-score is not a percentile or a probability; it is a count of standard deviations and can be negative.

Front: A test has mean 70 and standard deviation 8. Maria scored 86. Compute her z-score and interpret it.
Back: z = (86 − 70) ÷ 8 = 16 ÷ 8 = 2. Maria's score is 2 standard deviations above the mean.
Extra: Why students miss this: The z-score is the standardized distance, not the raw point difference of 16.

Front: Anya scored z = 1.5 on a biology test and z = 0.8 on a chemistry test (each class roughly normal). On which test did she perform better relative to her class, and why does standardizing let you compare?
Back: She did better in biology, because z = 1.5 places her further above her class mean than z = 0.8 does. Standardizing converts both scores to the same standard-deviation scale, so they can be compared even though the tests differ.
Extra: Why students miss this: You cannot compare raw scores from two different tests directly; standardizing first puts them on a common scale.

Front: A runner's finish time is at the 90th percentile among all finishers. In plain words, what does that mean?
Back: About 90% of finishers had times less than or equal to hers. Her time is high in the ordered list — only about 10% of finishers had a time as large or larger.
Extra: Why students miss this: A percentile is a position in the data (percent at or below), not the percent of the value itself or a score out of 100.

Front: What is a simple random sample (SRS)?
Back: A sample chosen so that every possible group of the desired size has an equal chance of being selected — equivalently, every individual has an equal chance and selections are made without favoritism.
Extra: Why students miss this: An SRS is not just 'haphazard' selection; it requires a genuine chance mechanism giving all samples equal likelihood.

Q: Match the sampling method: a [...] sample splits the population into similar groups and samples within each, while a [...] sample splits into groups and randomly selects whole groups.
A: stratified; cluster
Extra: Stratified sampling samples within every group (stratum); cluster sampling randomly chooses entire groups (clusters) and uses everyone in the chosen groups.
Why students miss this: Stratified and cluster are opposites in approach: strata are sampled from individually, clusters are selected as whole units.

Front: Define systematic sampling.
Back: You order the population and select every kth individual after a random starting point (for example, every 10th person on a list).
Extra: Why students miss this: Systematic sampling is not the same as a simple random sample, though it can be reasonable if the list order is unrelated to the variable of interest.

Front: A website posts a poll and only people who feel strongly bother to respond. What kind of bias is this, and which direction can it push results?
Back: It is voluntary response bias. People with strong (often negative) opinions are overrepresented, so the results can be systematically skewed away from the true population view.
Extra: Why students miss this: A large number of voluntary responses does not fix the bias; the self-selection itself distorts the sample.

Front: A surveyor interviews only shoppers at one mall on a weekday morning to estimate a city's opinions. Identify the sampling problem and the resulting bias.
Back: This is a convenience sample, and it causes undercoverage bias: people who never visit that mall at that time (workers, students, those elsewhere) have little or no chance of being included, so the sample is not representative.
Extra: Why students miss this: Convenience sampling is biased regardless of how many people are interviewed; adding more shoppers does not make it representative.

Front: Why does using random selection to choose a sample help produce trustworthy conclusions about a population?
Back: Random selection removes human choice from who gets picked, so on average the sample resembles the population and no group is systematically favored. This reduces bias and lets you generalize from the sample.
Extra: Why students miss this: Randomness does not guarantee a perfect sample every time; it prevents systematic bias on average, which is what allows valid generalization.

Front: What is the key difference between an observational study and an experiment?
Back: In an observational study, researchers measure variables without intervening. In an experiment, researchers deliberately impose a treatment on subjects to see its effect. Only an experiment can support a cause-and-effect conclusion.
Extra: Why students miss this: An observational study cannot establish causation, no matter how strong the association it finds.

Q: The three pillars of a well-designed experiment are [...], [...], and [...].
A: control; randomization; replication
Extra: Control (hold other conditions similar and use a comparison group), randomization (randomly assign subjects to treatments), and replication (enough subjects, and repeatable results).
Why students miss this: Random assignment of treatments (experiment) is different from random selection of subjects (sampling); good experiments need control and replication too.

Front: A study finds people who drink more coffee sleep less, but heavy coffee drinkers also tend to work night shifts. Why does this make it hard to conclude coffee causes less sleep?
Back: Work schedule is a confounding variable: it is tied to both coffee intake and sleep, so its effect cannot be separated from coffee's. The observed link could be due to night shifts rather than coffee itself.
Extra: Why students miss this: An association in an observational study can be explained by a confounder, so it does not prove coffee causes reduced sleep.

Front: In a drug trial, neither the patients nor the staff measuring outcomes know who got the real pill versus a placebo. What is this design feature called, and why is it used?
Back: It is double-blinding. It prevents patients' and evaluators' expectations from influencing reported or measured outcomes, so the comparison reflects the treatment rather than the power of suggestion or observer bias.
Extra: Why students miss this: A placebo alone is not enough; if staff know the assignments, their expectations can still bias the results.

Front: What does random assignment of treatments allow you to conclude that random sampling does not, and vice versa?
Back: Random assignment (in an experiment) lets you draw cause-and-effect conclusions. Random sampling lets you generalize results to the larger population. You need both to both generalize and claim causation.
Extra: Why students miss this: Random sampling does not establish causation, and random assignment alone does not let you generalize beyond the subjects studied.

Front: Researchers randomly assign 60 volunteers who answered an online ad to two study-method treatments and find a difference. To whom can the cause-and-effect finding be generalized, and why?
Back: Because treatments were randomly assigned, a cause-and-effect conclusion is justified — but only for these 60 volunteers, since they were not randomly selected from a larger population. The result should not be generalized to all students.
Extra: Why students miss this: Self-selected volunteers limit generalization even when random assignment supports a causal claim within the study group.

Front: What does the probability of an outcome mean as a long-run idea?
Back: The proportion of times that outcome would happen if you repeated the random process a very large number of times.
Extra: Why students miss this: Probability does not predict the next single trial; a fair coin can still land heads five times in a row.

Front: A student flips a coin 8 times and gets 6 heads, then claims the coin is unfair. Why is this a weak conclusion?
Back: Eight flips is far too few for the proportion to settle near its long-run value, so a result like 6 heads is well within ordinary chance variation for a fair coin.
Extra: Why students miss this: Short runs are expected to be lumpy; only large numbers of trials make the relative frequency stabilize.

Front: You want to estimate the chance that a 4-player game ends in a tie, but the math is messy. How can a simulation give you an answer?
Back: Model one play of the game with random digits or a random number generator, repeat it many times, then use the proportion of simulated plays that ended in a tie as the estimated probability.
Extra: Why students miss this: A simulation estimates the probability from many imitated trials; it does not compute the exact theoretical value.

Q: Every legitimate probability must be a number between [...], and the probabilities of all outcomes in the sample space sum to [...].
A: 0 and 1; 1
Extra: Each probability lies from 0 to 1 inclusive, and the complete set of disjoint outcomes adds to exactly 1.
Why students miss this: A reported 'probability' above 1 or below 0 is impossible and signals an arithmetic error.

Front: In a two-way table, what is a marginal relative frequency?
Back: The proportion of the whole group falling in one category of a single variable — a row total or column total divided by the grand total.
Extra: Why students miss this: A marginal frequency uses the grand total as its denominator, ignoring the other variable entirely.

Front: What is a joint relative frequency in a two-way table?
Back: The proportion of the whole group that falls in one specific cell — that cell's count divided by the grand total — describing both variables at once.
Extra: Why students miss this: A joint frequency divides by the grand total, not by a single row or column total.

Front: A survey table cross-tabulates owning a pet (yes/no) with living in an apartment (yes/no). You want the proportion of apartment dwellers who own a pet. Which kind of relative frequency is that, and what is the denominator?
Back: A conditional relative frequency; the denominator is the total number of apartment dwellers (that column's total), not the grand total.
Extra: Why students miss this: Conditioning restricts you to one row or column total; using the grand total answers a different question.

Front: How can you use a two-way table to judge whether two categorical variables look associated?
Back: Compare the conditional distributions of one variable across the categories of the other; if those distributions differ noticeably, the variables appear associated.
Extra: Why students miss this: Equal raw counts do not show no association; you must compare proportions within each group.

Front: Of 200 people, 120 are students and 50 of those students subscribe to a streaming service. What proportion of students subscribe, and what proportion of everyone is a subscribing student?
Back: Among students, 50/120 ≈ 0.417 subscribe (conditional). Of everyone, 50/200 = 0.25 are subscribing students (joint).
Extra: Why students miss this: The conditional and joint values differ because their denominators differ; do not report the same number for both.

Front: State the complement rule for an event A.
Back: P(not A) = 1 − P(A); the chance an event does not happen is one minus the chance it does.
Extra: Why students miss this: The complement is 1 minus P(A), not the probability of some other named event.

Front: What does it mean for two events to be mutually exclusive (disjoint)?
Back: They cannot both occur on the same trial, so P(A and B) = 0.
Extra: Why students miss this: Mutually exclusive means they never overlap; it is not the same as being independent.

Front: Write the general addition rule for P(A or B) and say why the last term is there.
Back: P(A or B) = P(A) + P(B) − P(A and B); you subtract the overlap because outcomes in both A and B would otherwise be counted twice.
Extra: Why students miss this: You only drop the subtraction term when the events are mutually exclusive (overlap = 0).

Front: In a deck, P(face card) = 12/52 and P(heart) = 13/52, with 3 cards both face and heart. What is P(face or heart)?
Back: 12/52 + 13/52 − 3/52 = 22/52 ≈ 0.423, after subtracting the 3 cards counted in both groups.
Extra: Why students miss this: Forgetting to subtract the 3 overlap cards overstates the probability.

Front: A weather app says P(rain) = 0.30. A friend insists P(no rain) might be 0.80. Why can't that be right?
Back: Rain and no-rain are complements, so their probabilities must total 1; if P(rain) = 0.30 then P(no rain) must be 0.70, not 0.80.
Extra: Why students miss this: Complementary probabilities must sum to exactly 1, leaving no room for an inconsistent pair.

Front: Define conditional probability P(A | B) in words.
Back: The probability that A happens given that B has already happened — it restricts attention to the outcomes where B is true.
Extra: Why students miss this: P(A | B) and P(B | A) are different questions and usually have different values.

Front: Give the formula for P(A | B) using joint and marginal probabilities.
Back: P(A | B) = P(A and B) ÷ P(B); divide the chance both happen by the chance of the condition.
Extra: Why students miss this: The denominator is P(B), the event you conditioned on, not P(A) or the total.

Front: A test for a condition is positive in 90% of people who have it. A patient asks, 'Given my positive test, what's the chance I actually have it?' Why isn't that 90%?
Back: 90% is P(positive | has it), but the patient wants P(has it | positive); reversing a conditional probability also depends on how common the condition is, so the two values are generally different.
Extra: Why students miss this: P(A | B) is not interchangeable with P(B | A); confusing them is the classic conditional-probability reversal error.

Front: Of 80 commuters, 30 take the train and 18 of those train riders are late. Estimate P(late | takes train).
Back: 18/30 = 0.60 — among train riders only, 60% are late.
Extra: Why students miss this: Dividing 18 by 80 ignores the condition and answers the wrong question.

Front: Write the general multiplication rule for P(A and B).
Back: P(A and B) = P(A) × P(B | A); the chance both happen is the chance of the first times the chance of the second given the first.
Extra: Why students miss this: You may only replace P(B | A) with P(B) when A and B are independent.

Front: A box has 5 red and 3 blue chips. You draw two without replacement. What is the probability both are red?
Back: P(both red) = (5/8) × (4/7) = 20/56 ≈ 0.357, since after the first red there are 4 reds left out of 7 chips.
Extra: Why students miss this: Without replacement the second probability changes; multiplying 5/8 by 5/8 wrongly assumes replacement.

Front: What does it mean for events A and B to be independent?
Back: Knowing whether B happened does not change the probability of A — that is, P(A | B) = P(A).
Extra: Why students miss this: Independent is about probabilities being unchanged, not about events being unable to occur together (that is mutually exclusive).

Front: Give the multiplication check for independence of A and B.
Back: A and B are independent exactly when P(A and B) = P(A) × P(B).
Extra: Why students miss this: This shortcut only holds under independence; in general you need the conditional term P(B | A).

Front: Why can two events that are mutually exclusive (and each have positive probability) never be independent?
Back: If they are mutually exclusive, learning that one occurred tells you the other definitely did not, so P(A | B) drops to 0 — that is a change from P(A), which breaks independence.
Extra: Why students miss this: Mutually exclusive events are strongly dependent, the opposite of independent.

Front: A machine's two parts fail independently with P = 0.02 each. What is the probability that at least one part fails?
Back: Use the complement: 1 − P(neither fails) = 1 − (0.98)(0.98) = 1 − 0.9604 = 0.0396.
Extra: Why students miss this: Adding 0.02 + 0.02 double-counts the case where both fail; the complement avoids that.

Front: A gambler says, 'The roulette wheel landed red five times, so black is due.' What error is this?
Back: Spins are independent, so past results do not change the next spin's probability; the wheel has no memory. This is the gambler's fallacy.
Extra: Why students miss this: Independent trials never become 'due'; the next outcome's probability is unchanged by the streak.

Front: What is a random variable?
Back: A variable that assigns a number to each outcome of a random process, so its value depends on chance.
Extra: Why students miss this: A random variable takes numeric values; a categorical outcome like 'red' is not itself a random variable until coded as a number.

Front: What makes a variable a discrete random variable rather than continuous?
Back: Its possible values are countable and separate (like 0, 1, 2, ...), rather than filling an entire interval of the number line.
Extra: Why students miss this: Discrete values are countable; whole-number counts are typical, while measurements like time tend to be continuous.

Front: What two conditions must a valid discrete probability distribution satisfy?
Back: Every listed probability is between 0 and 1, and all of the probabilities add up to exactly 1.
Extra: Why students miss this: A table whose probabilities do not sum to 1 is not a valid distribution, no matter how reasonable each entry looks.

Front: A distribution lists P(0)=0.2, P(1)=0.3, P(2)=0.4, P(3)=?. What must P(3) be, and why?
Back: P(3) = 0.1, because the probabilities of all outcomes must total 1, and 0.2 + 0.3 + 0.4 = 0.9 leaves 0.1.
Extra: Why students miss this: You cannot pick P(3) freely; it is forced by the requirement that the total equals 1.

Front: Using a probability distribution, how do you find P(X is at least 2) for a discrete variable with values 0,1,2,3?
Back: Add the probabilities of all values that are 2 or more: P(2) + P(3); 'at least 2' includes 2 itself.
Extra: Why students miss this: 'At least 2' is not P(X = 2) alone, and it does include the value 2.

Front: What does the mean (expected value) μ of a discrete random variable represent?
Back: The long-run average value of the variable over many repetitions — a probability-weighted center of the distribution.
Extra: Why students miss this: The expected value need not be a possible outcome; an average family size of 2.4 children is fine.

Front: How do you compute the mean μ of a discrete random variable from its distribution?
Back: Multiply each value by its probability and add the products: μ = sum of (value × its probability).
Extra: Why students miss this: It is a weighted average using the probabilities, not the plain average of the listed values.

Front: A raffle ticket wins $0 with probability 0.9, $5 with 0.08, and $50 with 0.02. What is the expected winning per ticket?
Back: μ = 0(0.9) + 5(0.08) + 50(0.02) = 0 + 0.40 + 1.00 = $1.40 per ticket on average.
Extra: Why students miss this: The expected value averages over all tickets; it is not the most likely single payout (which here is $0).

Front: A game costs $2 to play and the expected winning (before cost) is $1.40 per play. Should a player expect to profit, and why?
Back: No; the expected net result is $1.40 − $2.00 = −$0.60 per play, so over many plays the player loses about 60 cents each time on average.
Extra: Why students miss this: A negative expected net means a long-run loss even though an individual play can still win big.

Front: What does the standard deviation σ of a discrete random variable describe?
Back: The typical distance of the variable's value from its mean — how spread out the outcomes are around μ.
Extra: Why students miss this: Standard deviation measures spread, not the center; a large μ does not imply a large σ.

Front: How is the standard deviation of a discrete random variable obtained from its variance?
Back: Compute the variance as the probability-weighted average of the squared distances from μ, then take its square root to get σ.
Extra: Why students miss this: Standard deviation is the square root of the variance, not the variance itself; its units match the variable.

Front: Two delivery services both average 3 days, but service A has σ = 0.4 days and service B has σ = 1.8 days. Which is more reliable, and why?
Back: Service A; the smaller standard deviation means its delivery times cluster tightly near 3 days, so it is more predictable, while B varies widely.
Extra: Why students miss this: Equal means do not mean equal reliability; spread (σ) is what governs consistency.

Front: In a binomial setting, what exactly does the random variable count?
Back: The number of successes in the fixed number of trials.
Extra: Why students miss this: A binomial variable counts successes in a fixed n; it does not count trials until the first success (that removed setting is geometric).

Front: A quality inspector checks 12 randomly chosen bulbs and counts how many are defective, where each bulb is defective with probability 0.05 independently. Is this binomial, and what are n and p?
Back: Yes — fixed n = 12 trials, two outcomes (defective/not), constant p = 0.05, independent bulbs; the count of defectives is binomial with n = 12, p = 0.05.
Extra: Why students miss this: Random selection keeps trials effectively independent here; if you sampled a large fraction of a small batch without replacement, the binomial model would break down.

Q: For a binomial variable, the mean is [...] and the standard deviation is [...].
A: np; the square root of np(1 − p)
Extra: Mean = np; standard deviation = square root of np(1 − p), where n is the number of trials and p the success probability.
Why students miss this: The SD uses np(1 − p) under the square root, not np; do not confuse it with the mean.

Front: A free-throw shooter makes 80% of shots and takes 25 shots. What are the expected number made and the standard deviation?
Back: Mean = np = 25(0.8) = 20 made; SD = square root of 25(0.8)(0.2) = square root of 4 = 2 shots.
Extra: Why students miss this: Use p = 0.8 for success and 1 − p = 0.2 in the SD; mixing them up changes the variance.

Front: Why does a binomial probability for 'exactly k successes' multiply by a count of the number of arrangements?
Back: Because the k successes can occur in many different orders among the n trials, and each arrangement has the same probability, so you add them all by multiplying one arrangement's probability by the number of possible orderings.
Extra: Why students miss this: Forgetting the number of arrangements only gives the chance of one specific order, undercounting the true probability.

Front: A multiple-choice quiz has 10 questions, each guessed with p = 0.25 of being right. How would you find the probability of getting at least 8 correct?
Back: Add the binomial probabilities for exactly 8, exactly 9, and exactly 10 correct, since 'at least 8' covers those three counts.
Extra: Why students miss this: 'At least 8' is not just P(exactly 8); you must include the larger counts too.

Front: When is it reasonable to treat a count from sampling without replacement as approximately binomial?
Back: When the sample is no more than about 10% of the population, so removing items barely changes the success probability across trials.
Extra: Why students miss this: Sampling without replacement is only roughly binomial under the 10% condition; for large sample fractions the dependence matters.

Front: What does a z-score (standardized value) tell you?
Back: How many standard deviations a value is above or below the mean; a positive z is above the mean and a negative z is below.
Extra: Why students miss this: A z-score is in units of standard deviations, not in the variable's original units.

Front: How do you standardize a value x from a normal model with mean μ and standard deviation σ?
Back: z = (x − μ) ÷ σ; subtract the mean, then divide by the standard deviation.
Extra: Why students miss this: You divide by σ after subtracting μ; reversing the order or skipping the subtraction gives the wrong z.

Front: Heights are normal with mean 64 in and SD 3 in. Using the empirical rule, about what percent of people are between 61 and 67 inches?
Back: About 68%, because 61 and 67 are exactly one standard deviation below and above the mean of 64.
Extra: Why students miss this: The 68% figure applies only because the bounds are one SD out; other bounds need different percents or a z-table.

Front: Test A scores normal with mean 70, SD 5; test B with mean 80, SD 4. A student scores 78 on A and 86 on B. On which test did the student do relatively better?
Back: Test A: z = (78 − 70)/5 = 1.6 versus test B: z = (86 − 80)/4 = 1.5; the higher z-score on A means a better standing relative to peers, even though the raw score is lower.
Extra: Why students miss this: Higher raw scores do not always mean better relative performance; standardizing accounts for different means and spreads.

Front: A delivery time has mean 30 min and SD 6 min, but its distribution is strongly right-skewed. Why is the empirical rule a poor tool here?
Back: The 68–95–99.7 rule assumes a normal, symmetric bell shape; a skewed distribution does not match those proportions, so the percents would be misleading.
Extra: Why students miss this: The empirical rule is not a general fact about all data; it only fits approximately normal distributions.

Front: Battery life is normal with mean 500 h and SD 40 h. A battery lasts 580 h. Find its z-score and describe what it says.
Back: z = (580 − 500)/40 = 2.0, meaning this battery lasts 2 standard deviations above the mean — an unusually long-lasting battery.
Extra: Why students miss this: A z of 2.0 is notably large; treating 580 as 'just a bit above average' understates how unusual it is.

Front: What is the difference between a parameter and a point estimator?
Back: A parameter is the fixed (usually unknown) value that describes the whole population, such as the true proportion p. A point estimator is a statistic computed from a sample, such as p̂, that we use to estimate that parameter.
Extra: Why students miss this: p̂ is not the parameter — it changes from sample to sample, while p stays fixed.

Front: What does it mean for an estimator like p̂ to be unbiased?
Back: Over many random samples, the values of the estimator center on the true parameter — the mean of its sampling distribution equals the parameter it estimates. It does not systematically over- or under-shoot.
Extra: Why students miss this: Unbiased does not mean any single estimate equals the parameter; it is about the long-run average across samples.

Q: As sample size increases, the [...] of a point estimator [...], so estimates cluster more tightly around the parameter.
A: variability; decreases
Extra: Larger samples give estimators with less variability (smaller spread), so individual estimates land closer to the true parameter.

Front: Two methods estimate the same proportion. Method A is unbiased but has high variability; Method B has low variability but is biased high. Which property do you want, and what is the ideal?
Back: Ideally you want both low bias and low variability. Between these two, an unbiased estimator is generally preferred because it does not systematically miss the target — but the best estimator is one that is unbiased AND has small variability.
Extra: Why students miss this: Low variability alone is not enough — a tightly clustered estimator that is consistently off-target is still misleading.

Front: What is the sampling distribution of the sample proportion p̂?
Back: The distribution of the values p̂ would take across all possible random samples of a fixed size n from the population. It describes how p̂ varies from sample to sample.

Front: A population has true proportion p = 0.40. For samples of size n = 100, what are the mean and standard deviation of the sampling distribution of p̂?
Back: Mean = p = 0.40. Standard deviation = sqrt(p(1-p)/n) = sqrt(0.40 × 0.60 / 100) = sqrt(0.0024) ≈ 0.049.
Extra: Why students miss this: Do not divide by n inside without multiplying p(1-p) first, and do not forget the square root.

Front: Why is the center (mean) of the sampling distribution of p̂ equal to the true proportion p?
Back: Because p̂ is an unbiased estimator of p: across all possible random samples the values of p̂ average out to p, with no systematic tendency to be too high or too low.

Front: If you increase the sample size from n to 4n, what happens to the standard deviation of p̂?
Back: It is cut in half. The SD is sqrt(p(1-p)/n), so multiplying n by 4 divides the SD by sqrt(4) = 2.
Extra: Why students miss this: Quadrupling n does not quarter the SD — the square root means it only halves it.

Q: The Large Counts condition lets us treat the sampling distribution of p̂ as approximately normal when both [...] ≥ 10 and [...] ≥ 10.
A: np; n(1-p)
Extra: Both the expected number of successes (np) and the expected number of failures (n(1-p)) must be at least 10 for the normal approximation to be reasonable.

Front: A sample of n = 50 is taken where the proportion of interest is about p = 0.05. Can you model the sampling distribution of p̂ as approximately normal? Explain.
Back: No. Check Large Counts: np = 50 × 0.05 = 2.5, which is less than 10. Since the expected number of successes is below 10, the normal approximation is not appropriate here.
Extra: Why students miss this: A 'big' n alone is not enough — if p is extreme, np or n(1-p) can still fall below 10.

Front: Why do we check the 10% condition (sample no more than 10% of the population) when sampling without replacement?
Back: Sampling without replacement makes observations slightly dependent. Keeping the sample under 10% of the population keeps that dependence small enough that the usual standard deviation formula for p̂ stays accurate.
Extra: Why students miss this: The 10% condition is about the population size, not the sample being 'big enough' — that is the Large Counts condition.

Front: What is the general structure of a confidence interval?
Back: Point estimate ± margin of error, where the margin of error = (critical value) × (standard error of the estimate).

Front: In a random sample of 200 voters, 110 support a measure. Construct a 95% confidence interval for the true proportion of supporters.
Back: p̂ = 110/200 = 0.55. SE = sqrt(0.55 × 0.45 / 200) ≈ 0.0352. With z* = 1.96, margin = 1.96 × 0.0352 ≈ 0.069. Interval: 0.55 ± 0.069, or about (0.481, 0.619).
Extra: Why students miss this: Use p̂ in the standard error for a confidence interval (there is no hypothesized p here).

Front: Why does a one-proportion confidence interval use p̂ in the standard error, while a significance test uses p0?
Back: A confidence interval has no hypothesized value, so the only available estimate of p is p̂. A significance test assumes H0 is true, so it uses the hypothesized p0 to compute the standard deviation under that assumption.
Extra: Why students miss this: Mixing these up (using p0 in an interval or p̂ in the test statistic) is a common scoring error.

Front: Keeping the sample the same, what happens to the width of a confidence interval if you raise the confidence level from 90% to 99%?
Back: The interval gets wider. A higher confidence level needs a larger critical value (z*), which increases the margin of error and widens the interval.
Extra: Why students miss this: Higher confidence does not make the estimate more precise — it makes the interval wider, not narrower.

Front: You want a 95% confidence interval for a proportion with margin of error no more than 0.03, and have no prior estimate of p. What sample size should you use?
Back: Use the conservative guess p̂ = 0.5 (maximizes p̂(1-p̂)). Set 1.96 × sqrt(0.5 × 0.5 / n) ≤ 0.03. Solving: n ≥ (1.96 / 0.03)^2 × 0.25 ≈ 1067.1, so use n = 1068 (round up).
Extra: Why students miss this: Always round the required sample size UP, never down, or the margin of error will exceed your target.

Front: A study reports a 95% confidence interval of (0.48, 0.56) for the proportion of adults who exercise daily. Write a correct one-sentence interpretation.
Back: We are 95% confident that the true proportion of all adults who exercise daily is between 0.48 and 0.56.
Extra: Why students miss this: Do not say there is a 95% probability the proportion is in this specific interval — the parameter is fixed; the interval is what varies across samples.

Front: What does the confidence level (e.g. 95%) actually describe?
Back: It describes the method: if we took many random samples and built an interval from each the same way, about 95% of those intervals would capture the true parameter. It is a property of the procedure, not of any single interval.
Extra: Why students miss this: 95% is not the chance the parameter falls in your one interval — it is the long-run capture rate of the method.

Front: A student writes: 'There is a 95% chance the true proportion lies between 0.40 and 0.50.' Why is this interpretation wrong?
Back: Once the interval is computed, the true proportion either is or is not inside it — there is no probability about the fixed parameter. The 95% refers to how often the method captures the parameter over many samples, not the probability for this one interval.
Extra: Why students miss this: Assigning a probability to the parameter being in a specific computed interval is the classic confidence-interval error.

Front: What is the difference between the null hypothesis (H0) and the alternative hypothesis (Ha)?
Back: H0 is the 'no effect / no difference' claim taken as true to start, usually an equality (p = p0). Ha is what we are gathering evidence for — that the parameter is greater than, less than, or different from the null value.

Front: A company claims more than 30% of customers reorder. You want to test this. Write H0 and Ha in symbols, defining p.
Back: Let p = the true proportion of all customers who reorder. H0: p = 0.30 versus Ha: p > 0.30.
Extra: Why students miss this: Hypotheses are written about the parameter p, not the sample proportion p̂.

Front: Why must hypotheses be stated in terms of the population parameter p and never the sample statistic p̂?
Back: A test is about an unknown truth in the population. We already know p̂ from the data, so claims about it would be either trivially true or false. The whole point is to draw a conclusion about the unknown population value p.
Extra: Why students miss this: Writing H0: p̂ = 0.30 is incorrect; the statistic is data, not a hypothesis.

Q: The three conditions for a one-proportion z test are: data from [...], the [...] condition (sampling without replacement), and Large Counts using [...].
A: a random sample/randomization; 10%; p0
Extra: Random sample or random assignment; sample no more than 10% of the population; and n×p0 ≥ 10 and n×(1-p0) ≥ 10 (Large Counts checked with the hypothesized p0).

Front: When checking Large Counts and computing the standard error for a one-proportion test, why do you use p0 rather than p̂?
Back: A significance test works by assuming H0 is true. Under that assumption the true proportion is p0, so the standard deviation of p̂ is computed with p0. Conditions and the test statistic are both evaluated 'as if H0 holds.'
Extra: Why students miss this: Using p̂ in the test statistic's standard error (instead of p0) is a frequent mistake — that form belongs to confidence intervals.

Front: Define a p-value.
Back: The probability of getting a sample result at least as extreme as the one observed, in the direction of Ha, assuming the null hypothesis is true.
Extra: Why students miss this: The p-value is not the probability that H0 is true — it is the probability of the data (or more extreme) given H0.

Front: Test H0: p = 0.50 vs Ha: p > 0.50. A sample of n = 400 gives p̂ = 0.55. Compute the z test statistic.
Back: Standard error under H0 = sqrt(0.50 × 0.50 / 400) = sqrt(0.000625) = 0.025. z = (0.55 - 0.50) / 0.025 = 2.0.
Extra: Why students miss this: Use p0 = 0.50 (not p̂ = 0.55) in the standard error of a test statistic.

Front: A test gives a p-value of 0.03. Interpret this value in plain words for the context of testing whether a coin favors heads.
Back: If the coin were actually fair (H0 true), there would be only a 3% chance of getting a result at least as favorable to heads as the one observed. Such a result is fairly unlikely under 'fair', giving evidence the coin favors heads.
Extra: Why students miss this: 0.03 is not the probability the coin is fair; it is the probability of the observed (or more extreme) data IF it were fair.

Front: For the same z test statistic, how does the p-value of a two-sided test (Ha: p ≠ p0) compare to a one-sided test (Ha: p > p0)?
Back: The two-sided p-value is twice the one-sided p-value, because it counts extreme results in both tails rather than just one.
Extra: Why students miss this: Forgetting to double the tail probability for a two-sided alternative understates the p-value.

Front: A test of H0: p = 0.20 vs Ha: p > 0.20 yields a p-value of 0.012 at α = 0.05. State the conclusion in context.
Back: Since 0.012 < 0.05, reject H0. There is convincing statistical evidence that the true proportion is greater than 0.20.
Extra: Why students miss this: Rejecting H0 supports Ha; it does not 'prove' Ha is true.

Front: Why do we say 'fail to reject H0' instead of 'accept H0'?
Back: A large p-value means the data are consistent with H0, but consistency is not proof. Other parameter values may also be consistent with the data, so we lack evidence against H0 rather than evidence that it is exactly true.
Extra: Why students miss this: Failing to reject H0 never proves H0 true — it just means there was not enough evidence against it.

Front: A 95% confidence interval for p is (0.52, 0.61). Without computing, what would a two-sided test of H0: p = 0.50 at α = 0.05 conclude?
Back: Reject H0. Since 0.50 is not inside the 95% interval, a two-sided test at α = 0.05 would find significant evidence that p ≠ 0.50. (A 95% interval and a two-sided 0.05 test agree.)
Extra: Why students miss this: The CI–test link holds only when the confidence level and test match (95% interval with two-sided α = 0.05).

Front: Define a Type I error.
Back: Rejecting the null hypothesis when it is actually true — concluding there is an effect when there really is none (a false alarm).

Front: Define a Type II error.
Back: Failing to reject the null hypothesis when it is actually false — missing a real effect that is truly present.

Front: A smoke detector test has H0: no fire. The alarm goes off (rejects H0) but there is no fire. Which type of error is this, and why?
Back: A Type I error. H0 ('no fire') is true, but it was rejected — a false alarm. Rejecting a true null is by definition a Type I error.
Extra: Why students miss this: Match the error to (a) whether H0 is true and (b) whether you rejected it — not to which outcome 'feels worse.'

Front: How is the significance level α related to the probability of a Type I error?
Back: α IS the probability of a Type I error: when H0 is true, you reject it about α of the time. Choosing a smaller α lowers the chance of a false alarm.
Extra: Why students miss this: Lowering α reduces Type I error but raises Type II error (and lowers power) — there is a trade-off.

Front: Define the power of a significance test.
Back: The probability that the test correctly rejects the null hypothesis when a specific alternative is true. Power = 1 − P(Type II error).

Front: Name three ways to increase the power of a test (other than the true effect being larger).
Back: Increase the sample size; increase the significance level α (accept more Type I risk); and reduce variability (e.g., better measurement or design). A larger true effect also raises power but is not something you control.
Extra: Why students miss this: Increasing power by raising α also increases the Type I error rate — the two move together.

Front: A drug is tested with H0: the drug is no better than placebo. Describe the real-world consequence of a Type II error here.
Back: A Type II error means failing to detect that the drug actually works — concluding it is no better than placebo when it truly is better. The consequence is that an effective drug gets shelved and patients miss out on a real benefit.
Extra: Why students miss this: Do not confuse this with Type I (falsely approving an ineffective drug); a Type II error is missing a real effect.

Front: When comparing two groups, what parameter does a two-proportion procedure estimate or test?
Back: The difference between the two population proportions, p1 − p2.

Front: Group A: 60 of 150 succeed. Group B: 45 of 150 succeed. Find the standard error used for a confidence interval for p1 − p2.
Back: p̂1 = 0.40, p̂2 = 0.30. SE = sqrt(0.40×0.60/150 + 0.30×0.70/150) = sqrt(0.0016 + 0.0014) = sqrt(0.0030) ≈ 0.0548.
Extra: Why students miss this: For an interval, use each group's own p̂ in its own term — do NOT pool (pooling is only for the test).

Front: A 95% confidence interval for p1 − p2 (treatment minus control) is (−0.02, 0.11). What does this tell you about a difference between the groups?
Back: Because the interval includes 0, it is plausible the two proportions are equal — there is not convincing evidence of a real difference at the 95% level. The data are consistent with anything from a small decrease to a moderate increase.
Extra: Why students miss this: Containing 0 means 'no significant difference,' not that the proportions are definitely equal.

Front: Why does a two-proportion z test use a pooled (combined) proportion in the standard error?
Back: The null hypothesis says the two proportions are equal (p1 = p2), so under H0 there is one common proportion. Pooling all successes over all observations gives the best single estimate of that common value, used to compute the standard error as if H0 is true.
Extra: Why students miss this: Pooling is for the test only; a confidence interval keeps the two p̂ values separate.

Front: Group 1: 80 of 200 successes. Group 2: 60 of 200 successes. Compute the pooled proportion and the test statistic z for H0: p1 = p2.
Back: Pooled p̂ = (80+60)/(200+200) = 140/400 = 0.35. SE = sqrt(0.35×0.65×(1/200 + 1/200)) = sqrt(0.2275 × 0.01) ≈ 0.0477. z = (0.40 − 0.30)/0.0477 ≈ 2.10.
Extra: Why students miss this: Use the pooled p̂c in BOTH variance terms of the test's standard error.

Front: Researchers ask whether a smaller proportion of online shoppers return items than in-store shoppers. Define the parameters and write H0 and Ha.
Back: Let p1 = true return proportion for online shoppers and p2 = true return proportion for in-store shoppers. H0: p1 = p2 (equivalently p1 − p2 = 0) vs Ha: p1 < p2.
Extra: Why students miss this: Be consistent about which group is p1; the direction of Ha must match that labeling.

Front: How do you decide between a chi-square test of homogeneity and a chi-square test of independence?
Back: Look at the sampling design. Homogeneity: separate random samples from two or more populations (or groups) are compared on one categorical variable. Independence: one single sample is classified by two categorical variables to see if they are associated.
Extra: Why students miss this: The arithmetic is identical; the difference is the study design and how the hypotheses are worded.

Front: In a chi-square test, a cell has row total 120, column total 90, and the table grand total is 300. Find the expected count for that cell.
Back: Expected count = (row total × column total) / grand total = (120 × 90) / 300 = 10800 / 300 = 36.
Extra: Why students miss this: Expected counts come from the marginal totals, not from the observed value in that cell.

Q: For a chi-square test on a two-way table, the degrees of freedom equal [...] × [...].
A: (number of rows − 1); (number of columns − 1)
Extra: df = (r − 1)(c − 1), where r is the number of rows and c is the number of columns in the table.

Front: State the hypotheses in words for a chi-square test of independence between favorite genre and age group within one sample.
Back: H0: favorite genre and age group are independent (not associated) in the population. Ha: favorite genre and age group are associated (not independent).
Extra: Why students miss this: Chi-square hypotheses are stated in words about the categorical variables, not as equations about a single proportion.

Front: A cell has observed count 30 and expected count 24. Compute that cell's contribution to the chi-square statistic.
Back: Contribution = (observed − expected)^2 / expected = (30 − 24)^2 / 24 = 36 / 24 = 1.5. The full statistic is the sum of these contributions over all cells.
Extra: Why students miss this: Divide by the expected count (not the observed), and square the difference before dividing.

Front: Before carrying out a chi-square test, what condition must the expected counts satisfy, and why expected (not observed)?
Back: All expected counts must be at least 5. This keeps the chi-square statistic's distribution well approximated by the chi-square model. We check expected counts because the reference distribution is built on what we'd expect under H0, not on the observed data.
Extra: Why students miss this: The 'at least 5' rule applies to expected counts, not observed counts.

Front: A chi-square test of homogeneity comparing exercise habits across three cities gives χ² = 14.2, df = 4, p-value = 0.007. State the conclusion in context at α = 0.05.
Back: Since 0.007 < 0.05, reject H0. There is convincing evidence that the distribution of exercise habits is not the same across the three cities.
Extra: Why students miss this: A homogeneity conclusion is about whether the distributions differ across groups, not about a single proportion or about causation.

Front: Why is a chi-square test always right-tailed, and what does a large χ² value indicate?
Back: Each term (O − E)^2 / E is non-negative, so the statistic is large only when observed counts differ a lot from what H0 predicts. A large χ² (small p-value) means big discrepancies from the null model — evidence against H0. Only the upper tail counts as 'extreme.'
Extra: Why students miss this: There is no left tail to consider — small χ² just means the data closely match H0.

Front: You have one categorical variable measured on two independent random samples and want to compare the proportion in one category. Should you use a two-proportion z test or a chi-square test, and does it matter?
Back: Either works when there are exactly two groups and two outcome categories. A two-proportion z test (two-sided) and a chi-square test of homogeneity give equivalent results (z² = χ²). Use chi-square when there are more than two groups or more than two categories.
Extra: Why students miss this: Chi-square is the go-to once any dimension exceeds 2; the two-proportion z test only handles a single 2-by-2 comparison.

Front: Two students each take a random sample of 40 students from the same school and compute the mean daily screen time. Why are their two sample means almost certainly different?
Back: Because of sampling variability: each random sample contains different individuals, so each x̄ varies from sample to sample even though both estimate the same population mean μ.
Extra: Why students miss this: Different sample means do not mean someone made an error — variation between samples is expected and predictable.

Front: Define the sampling distribution of the sample mean x̄.
Back: The distribution of the values x̄ takes over all possible random samples of a fixed size n drawn from the same population.
Extra: Why students miss this: It is the distribution of the sample means across many samples — not the distribution of individual data values in one sample.

Front: A population has mean μ = 70 and standard deviation σ = 12. For random samples of size n = 36, what are the mean and standard deviation of the sampling distribution of x̄?
Back: Mean of x̄ is μ = 70. Standard deviation of x̄ is σ/√n = 12/√36 = 12/6 = 2.
Extra: Why students miss this: The standard deviation of x̄ is σ/√n, not σ — averaging shrinks the spread.

Front: Why is the standard deviation of x̄ (σ/√n) smaller than the population standard deviation σ?
Back: Averaging several values lets highs and lows cancel out, so means cluster more tightly around μ than individual observations do. Dividing by √n captures this shrinkage.
Extra: Why students miss this: Increasing n shrinks the spread of x̄, not the spread of the underlying population.

Front: State what the Central Limit Theorem (CLT) tells us about the sampling distribution of x̄.
Back: For a large enough sample size, the sampling distribution of x̄ is approximately normal regardless of the population's shape. A common rule of thumb is n at least 30.
Extra: Why students miss this: The CLT is about the distribution of x̄, not about the population becoming normal or the sample data becoming normal.

Front: A population is strongly right-skewed. Under what conditions can we still treat the sampling distribution of x̄ as approximately normal?
Back: When the sample size is large enough (commonly n at least 30) so the Central Limit Theorem applies. If the population were already normal, any sample size would work.
Extra: Why students miss this: A skewed population does not forbid normality of x̄ — a large n fixes it via the CLT.

Front: A quality team can sample either n = 25 or n = 100 widgets to estimate mean weight. How does the larger sample change the sampling distribution of x̄?
Back: The larger sample makes the sampling distribution of x̄ narrower (smaller σ/√n) and closer to normal, so x̄ is a more precise estimate of μ.
Extra: Why students miss this: A bigger sample does not change the center (still μ) — it reduces the spread of x̄.

Q: The Central Limit Theorem says that as the sample size grows, the sampling distribution of x̄ becomes approximately [...] regardless of the shape of the [...].
A: normal; population
Extra: It becomes approximately normal regardless of the shape of the population.
Why students miss this: The approximation is for the distribution of x̄, not for the individual data values.

Front: When constructing a confidence interval for a population mean, why do we use a t-distribution instead of the standard normal (z)?
Back: Because the population standard deviation σ is unknown and we estimate it with the sample standard deviation s. That extra estimation adds uncertainty, which the t-distribution accounts for.
Extra: Why students miss this: We use t whenever σ is unknown — not only with small samples; the issue is estimating σ with s.

Front: For one-sample inference about a mean with sample size n, what are the degrees of freedom?
Back: df = n − 1.
Extra: Why students miss this: Degrees of freedom for one mean is n − 1, not n.

Front: How does the shape of a t-distribution compare to the standard normal curve?
Back: A t-distribution is bell-shaped and centered at 0 like the normal, but it has more spread and heavier tails. Smaller degrees of freedom give heavier tails.
Extra: Why students miss this: The t-curve is wider than z, so it gives a larger critical value t* and a wider interval for the same confidence level.

Front: What happens to the t-distribution as the degrees of freedom get very large?
Back: It approaches the standard normal (z) distribution, because a large sample estimates σ well, so s is close to σ.
Extra: Why students miss this: t never becomes exactly z for finite n, but it gets very close as df grows.

Front: Write the formula (in words and symbols) for a one-sample confidence interval for a population mean.
Back: x̄ ± t* · (s/√n): the sample mean plus or minus the critical value t* times the standard error s/√n. Use df = n − 1 to find t*.
Extra: Why students miss this: The standard error here is s/√n, and the multiplier is t* (not z*).

Front: A researcher wants a t-interval for the mean commute time from a random sample of 28 workers. What conditions must be checked?
Back: Random: data come from a random sample or randomized experiment. Normal/large: the population is roughly normal or n is large enough (CLT), checked with a graph showing no strong skew/outliers. Independence: sampling with replacement or the sample is at most 10% of the population.
Extra: Why students miss this: All three conditions must be checked — randomness alone is not enough to justify a t-interval.

Front: A sample of 25 batteries gives x̄ = 41.2 hours and s = 3.0 hours. With t* = 2.064 (df = 24), find the margin of error for a 95% confidence interval.
Back: Margin of error = t* · s/√n = 2.064 × (3.0/√25) = 2.064 × 0.6 = 1.24 hours.
Extra: Why students miss this: Divide s by √n (here √25 = 5), not by n, before multiplying by t*.

Front: A 95% confidence interval for mean fill volume is (498.2, 501.6) mL. Interpret this interval in context.
Back: We are 95% confident that the true mean fill volume of all bottles in the population is between 498.2 mL and 501.6 mL.
Extra: Why students miss this: The interval estimates the population mean μ, not an individual bottle and not the sample mean.

Front: What does the phrase '95% confidence' actually describe about the method?
Back: If we repeated the sampling and interval-building process many times, about 95% of the resulting intervals would capture the true population mean. It describes the long-run success rate of the method.
Extra: Why students miss this: It is wrong to say there is a 95% probability that μ is in this one interval — μ is fixed; the confidence is in the method.

Front: A label claims a mean weight of 50 g. A 95% confidence interval for the true mean is (51.0, 53.4) g. What can you conclude about the claim?
Back: Since 50 g is not inside the interval, the data give convincing evidence against the claim — the true mean appears to be higher than 50 g.
Extra: Why students miss this: A value outside the interval is implausible; a value inside would mean the data are consistent with the claim.

Front: Keeping the sample the same, how does raising the confidence level from 90% to 99% change the interval for a mean?
Back: It makes the interval wider, because a higher confidence level uses a larger t* (you must capture μ more often, so you cast a wider net).
Extra: Why students miss this: Higher confidence does not make the estimate more accurate — it just widens the interval; precision comes from larger n.

Front: At a fixed confidence level, how does increasing the sample size n affect the width of a confidence interval for a mean?
Back: It makes the interval narrower, because the standard error s/√n shrinks as n grows, reducing the margin of error.
Extra: Why students miss this: To halve the margin of error you must roughly quadruple n (because of the √n), not just double it.

Front: A company claims its light bulbs last on average 1000 hours. A consumer group suspects they last less. Write the hypotheses for a test about the mean μ.
Back: H0: μ = 1000 hours versus Ha: μ < 1000 hours, where μ is the true mean lifetime of all the bulbs.
Extra: Why students miss this: Hypotheses are statements about the population mean μ, never about the sample mean x̄.

Front: In a test for a mean, what does the null hypothesis H0 generally state?
Back: That the population mean equals a specific claimed value (no effect or no difference from the stated value), written H0: μ = μ0.
Extra: Why students miss this: H0 always uses equality (μ = μ0); the inequality goes in the alternative Ha.

Front: How do you decide whether the alternative hypothesis Ha for a mean should be one-sided (< or >) or two-sided (≠)?
Back: Use the wording before seeing the data: a directional claim ('more than', 'less than', 'decreased') gives a one-sided Ha; a claim that the mean simply 'differs' or 'changed' gives a two-sided ≠ Ha.
Extra: Why students miss this: You must not pick the direction of Ha after peeking at the data — that inflates the error rate.

Front: Write the formula for the one-sample t test statistic for a mean and say what it measures.
Back: t = (x̄ − μ0)/(s/√n). It measures how many estimated standard errors the sample mean falls from the hypothesized mean μ0.
Extra: Why students miss this: The denominator is the standard error s/√n, not just s.

Front: What does a p-value mean in a significance test for a mean?
Back: Assuming H0 is true, it is the probability of getting a sample mean at least as extreme as the one observed, in the direction(s) of Ha.
Extra: Why students miss this: A p-value is not the probability that H0 is true — it is computed assuming H0 is true.

Front: A test of a single mean uses a random sample of 18 trees with no strong skew or outliers in the dotplot. Which conditions are satisfied for a one-sample t test?
Back: Random (random sample), Normal/Large (n is small but the graph shows roughly symmetric data with no outliers, so the t procedure is reasonable), and Independent (sample is at most 10% of all trees).
Extra: Why students miss this: With a small n you must check a graph for skew/outliers — you cannot rely on the CLT alone.

Front: A test for a mean gives p = 0.018 at significance level α = 0.05. State the decision and what it means.
Back: Since 0.018 < 0.05, reject H0. There is convincing statistical evidence for the alternative claim about the mean.
Extra: Why students miss this: Compare p to α: small p means reject H0; it does not 'prove' Ha is true.

Front: Why do we say 'fail to reject H0' rather than 'accept H0' when the p-value is large?
Back: A large p-value means the data are consistent with H0, but it does not prove H0 is true — there could be a real effect the sample was too small to detect.
Extra: Why students miss this: Failing to reject H0 never confirms H0; it only means there isn't enough evidence against it.

Front: A test of H0: μ = 8 vs Ha: μ > 8 yields p = 0.21 at α = 0.05. Write a conclusion in context (μ is mean hours of sleep).
Back: Since 0.21 > 0.05, we fail to reject H0. There is not convincing evidence that the true mean number of hours of sleep is greater than 8.
Extra: Why students miss this: Always state the conclusion in context and link it to Ha, not just 'reject/fail to reject'.

Front: In a test about a mean, distinguish a Type I error from a Type II error.
Back: Type I: rejecting H0 when H0 is actually true (a false alarm). Type II: failing to reject H0 when Ha is actually true (a missed effect).
Extra: Why students miss this: The probability of a Type I error is α; it is not the p-value of a single test.

Front: Write the formula (in words and symbols) for a confidence interval for the difference of two means (independent samples).
Back: (x̄1 − x̄2) ± t* · √(s1²/n1 + s2²/n2): the difference in sample means plus or minus t* times the standard error of the difference.
Extra: Why students miss this: Standard errors add as variances: use √(s1²/n1 + s2²/n2), you cannot just add s1/√n1 and s2/√n2.

Front: A study compares mean test scores of two separately, randomly chosen classes. What conditions must hold for a two-sample t interval for the difference of means?
Back: Random: both samples are random (or from a randomized experiment). Normal/Large: each population is roughly normal or each sample is large enough. Independent: the two samples are independent of each other and each is at most 10% of its population.
Extra: Why students miss this: The two groups must be independent of each other — if they were paired, a two-sample procedure would be wrong.

Front: Why is the standard error of (x̄1 − x̄2) found by adding the two variances rather than subtracting them?
Back: Variances of independent quantities always add when you take a difference, because both samples contribute uncertainty. So SE = √(s1²/n1 + s2²/n2).
Extra: Why students miss this: Even though you subtract the means, the variabilities add — never subtract the variances.

Front: A 95% confidence interval for (μ_treatment − μ_control) is (1.2, 4.8) points. What does this say about the difference in means?
Back: Since the entire interval is above 0, we are 95% confident the treatment mean is higher than the control mean by between 1.2 and 4.8 points — convincing evidence of a real difference.
Extra: Why students miss this: An interval entirely above 0 indicates a real difference; an interval containing 0 would not.

Front: Why does a confidence interval for the difference of two means that contains 0 suggest no significant difference?
Back: Because 0 represents 'no difference' between the means. If 0 is a plausible value in the interval, the data are consistent with the two population means being equal.
Extra: Why students miss this: Containing 0 does not prove the means are equal — it just means a difference of 0 is plausible.

Front: Researchers compare the mean reaction times of two independent groups (caffeine vs none). Write the hypotheses for a test of the difference of means.
Back: H0: μ1 − μ2 = 0 (the population means are equal) versus Ha: μ1 − μ2 ≠ 0 (or a one-sided <, > if a direction is claimed), where μ1 and μ2 are the true mean reaction times.
Extra: Why students miss this: Hypotheses are about the difference of population means, written as μ1 − μ2 = 0, not about x̄1 − x̄2.

Front: Write the two-sample t test statistic for the difference of means.
Back: t = (x̄1 − x̄2 − 0)/√(s1²/n1 + s2²/n2): the observed difference in sample means divided by the standard error of that difference.
Extra: Why students miss this: The hypothesized difference (0 under H0) is subtracted in the numerator; the denominator combines both variances.

Front: A study measures each subject's blood pressure before and after a program and analyzes the change. Is this a two-sample or a matched-pairs design, and why?
Back: Matched pairs: the two measurements come from the same subject, so they are not independent. You analyze the single list of differences (after − before).
Extra: Why students miss this: Two measurements on the same individual are dependent, so a two-sample t test would be wrong — use paired t.

Front: Two separate, independently chosen groups of patients receive drug A or drug B, and their mean recovery times are compared. Two-sample or matched pairs?
Back: Two-sample: the two groups are made of different, independent individuals with no natural pairing, so you compare two independent means.
Extra: Why students miss this: Comparing two unrelated groups is a two-sample problem; only use paired methods when each value in one group is linked to a specific value in the other.

Front: What is the single biggest clue that data should be analyzed with a matched-pairs t procedure rather than a two-sample procedure?
Back: Each observation in one condition is naturally linked to a specific observation in the other (same subject measured twice, or subjects matched into pairs), so you work with one list of differences.
Extra: Why students miss this: Equal sample sizes do not signal pairing — pairing requires a meaningful link between each pair of values.

Front: In a matched-pairs test, what does μ_d represent, and what is the usual null hypothesis?
Back: μ_d is the true mean of the differences in the population of pairs. The usual null is H0: μ_d = 0 (no mean difference), tested against Ha: μ_d ≠ 0, < 0, or > 0.
Extra: Why students miss this: Paired analysis works with the mean of the differences (μ_d), not the difference of two separate means.

Front: Write the matched-pairs t test statistic for the mean difference.
Back: t = (x̄_d − 0)/(s_d/√n), where x̄_d is the mean of the differences, s_d is the standard deviation of the differences, and n is the number of pairs. Use df = n − 1.
Extra: Why students miss this: n is the number of pairs (differences), not the total number of measurements.

Front: Why does pairing (e.g., measuring the same subject twice) often make a study more able to detect a real effect?
Back: Pairing removes person-to-person variability, so the differences have a smaller standard deviation. A smaller s_d shrinks the standard error and makes a true mean difference easier to detect.
Extra: Why students miss this: Pairing reduces variability by controlling for differences between subjects; this is why a paired design can be more powerful than two independent samples.

Front: Twelve identical twin pairs are split so one twin gets a new diet and the other does not, then weight loss is compared within each pair. Which procedure applies?
Back: A matched-pairs t procedure: the twins form natural pairs, so analyze the difference in weight loss within each pair using one list of differences (n = 12 differences).
Extra: Why students miss this: Matching subjects into pairs before the study creates paired data, so a two-sample test is not appropriate.

Front: What conditions must be checked before running a matched-pairs t test?
Back: Random: pairs come from random sampling or random assignment within pairs. Normal/Large: the differences are roughly normal or the number of pairs is large enough, checked with a graph of the differences. Independent: pairs are independent and at most 10% of the population of pairs.
Extra: Why students miss this: The normality condition applies to the list of differences, not to the two original sets of measurements.

Front: A student runs a two-sample t test on before/after measurements taken from the same 30 people. What error did they make and what should they predict about the result?
Back: They wrongly treated dependent (paired) data as two independent samples. The correct analysis is a paired t test on the differences; ignoring the pairing inflates the standard error and can hide a real effect.
Extra: Why students miss this: Applying a two-sample test to paired data overstates the variability and weakens the test — pairing must be respected.

Front: A matched-pairs test of H0: μ_d = 0 vs Ha: μ_d > 0 gives t = 2.50 with 14 differences and p = 0.013 at α = 0.05. State the conclusion.
Back: Since 0.013 < 0.05, reject H0. There is convincing evidence that the true mean difference is greater than 0 (the 'after' values are higher on average).
Extra: Why students miss this: Tie the conclusion to μ_d and the context; df here is 14 − 1 = 13.

Front: A matched-pairs 95% confidence interval for μ_d (after − before reading speed) is (−1.5, 6.8) words per minute. What can you conclude?
Back: Because the interval contains 0, a mean difference of 0 is plausible, so there is not convincing evidence that the program changed mean reading speed.
Extra: Why students miss this: Containing 0 means 'no detected difference', not 'proof the program does nothing'.

Front: When is it appropriate to use a t procedure with the population standard deviation σ assumed known and use z instead?
Back: Essentially never in this course's mean inference: σ is treated as unknown, so we estimate it with s and use t. Use z only for proportions (where the SD comes from p), not for means.
Extra: Why students miss this: Means almost always use t (σ unknown); z procedures belong to proportion inference, not mean inference.

Front: Decide the procedure: a city measures air-quality index at 20 sites in the morning and again at the same 20 sites in the afternoon, comparing mean readings.
Back: Matched-pairs t procedure: each site is measured twice, so the two readings are paired by site. Analyze the 20 differences (afternoon − morning).
Extra: Why students miss this: Two measurements at the same locations are dependent; treating the 40 readings as two independent samples of 20 would be wrong.

Front: Decide the procedure: a researcher compares the mean income of a random sample of 50 college graduates with a separate random sample of 50 non-graduates.
Back: Two-sample t procedure for a difference of means: the two samples are independent and there is no natural pairing between a specific graduate and a specific non-graduate.
Extra: Why students miss this: Equal sample sizes (50 and 50) do not make data paired — pairing needs a one-to-one link, which is absent here.

Q: A paired-data analysis uses [...], so its test and interval are really just a [...] t procedure applied to those differences.
A: one list of differences; one-sample
Extra: It uses one list of differences, so it is a one-sample t procedure on the differences (df = n − 1, with n = number of pairs).
Why students miss this: Once you compute the differences, paired inference is identical to one-sample t inference — there is no separate 'paired formula' to memorize.

Front: In a study of two quantitative variables, which one goes on the x-axis: the explanatory or the response variable?
Back: The explanatory variable goes on the x-axis (horizontal); the response variable goes on the y-axis (vertical).
Extra: Why students miss this: The roles are not interchangeable for a graph — choosing which variable is explanatory reflects the question being asked, not just the data.

Front: A researcher wants to use hours studied to predict exam score. On the scatterplot, which variable belongs on the horizontal axis?
Back: Hours studied (the explanatory variable) goes on the horizontal axis; exam score (the response) goes on the vertical axis.
Extra: Why students miss this: Putting the variable of interest (exam score) on x just because it matters most is wrong — placement is set by which variable explains the other.

Q: When describing a scatterplot, name the four features to comment on: [...].
A: direction, form, strength, and unusual features (outliers)
Extra: Direction (positive/negative), form (linear/curved), strength (how tightly points cluster), and any unusual features such as outliers or clusters — always in context.
Why students miss this: Reporting only direction is incomplete; a full description always covers form, strength, and unusual points too.

Front: As outdoor temperature rises, monthly heating-fuel use falls. What is the direction of this association?
Back: Negative — as the explanatory variable increases, the response variable tends to decrease.
Extra: Why students miss this: Negative does not mean weak or bad; it only describes the downward trend of the points.

Front: A scatterplot of a bacterial population over time bends sharply upward and curves rather than following a straight path. How would you describe its form?
Back: Nonlinear (curved) — the points follow a curved pattern, so a straight-line model would not fit well.
Extra: Why students miss this: A strong, clear pattern is not automatically linear; strength and form are separate ideas.

Front: What does the strength of an association in a scatterplot describe?
Back: How closely the points follow the overall pattern — tightly clustered points mean a strong association, widely scattered points mean a weak one.
Extra: Why students miss this: Strength is about scatter around the pattern, not about how steep the trend is.

Front: Most points in a scatterplot lie along a tight upward band, but one point sits far above the rest at a low x-value. What unusual feature is this, and should you mention it?
Back: It is an outlier (a point that does not follow the overall pattern). Yes — unusual features must always be noted when describing a scatterplot.
Extra: Why students miss this: You should not ignore a single odd point; outliers can strongly affect later analysis.

Front: Ice-cream sales and drowning deaths both rise together across the year. Why is it wrong to say ice cream causes drownings?
Back: Association is not causation. A lurking variable — warm weather — drives both, so the two rise together without one causing the other.
Extra: Why students miss this: A strong association between two variables never by itself proves one causes the other.

Front: What does the correlation coefficient r measure?
Back: The direction and strength of the LINEAR association between two quantitative variables.
Extra: Why students miss this: r only measures linear association; it says nothing about curved relationships.

Q: The correlation r always falls between [...], inclusive.
A: -1 and 1
Extra: -1 and 1. Values near 1 or -1 indicate a strong linear pattern; values near 0 indicate a weak linear pattern.
Why students miss this: r can never be larger than 1 or smaller than -1; getting r = 1.4 means a calculation error.

Front: Why does the SIGN of r (positive or negative) matter?
Back: The sign gives the direction: positive r means the variables tend to increase together, negative r means one tends to decrease as the other increases.
Extra: Why students miss this: A negative r is not weaker than a positive one; r = -0.9 is just as strong as r = 0.9.

Front: A study reports r = 0.85 between weekly practice hours and free-throw percentage. Interpret this value.
Back: There is a strong, positive linear association: players who practice more tend to have higher free-throw percentages, and the points cluster fairly tightly around a line.
Extra: Why students miss this: r = 0.85 does not mean 85% of anything, and it does not prove practice causes higher percentages.

Front: A scatterplot shows a clear, strong U-shaped curve, yet r is close to 0. Why can that happen?
Back: Because r only captures LINEAR association. A strong curved pattern can have an r near 0 since the points do not follow a single straight line.
Extra: Why students miss this: r near 0 does NOT mean there is no relationship — it means no strong straight-line relationship.

Front: You recompute a correlation after converting heights from inches to centimeters. How does r change?
Back: It does not change. r has no units and is unaffected by changing the units or scale of either variable.
Extra: Why students miss this: Rescaling a variable does not strengthen or weaken r; correlation is unitless.

Front: If you swap which variable is treated as x and which is y, why does r stay the same?
Back: r is symmetric in the two variables — it measures their joint linear association, not a direction of prediction, so swapping x and y leaves r unchanged.
Extra: Why students miss this: Unlike the regression line, r does not depend on which variable is explanatory.

Front: A classmate says "r = 0.1, so there is no relationship between the variables." Why might that conclusion be wrong?
Back: A small r only rules out a strong straight-line pattern. The variables could still have a strong nonlinear (e.g. curved) relationship, so you must look at the scatterplot, not just r.
Extra: Why students miss this: Never judge a relationship from r alone — always view the scatterplot.

Front: Across many countries, number of cell phones per person and life expectancy have r = 0.78. Can we conclude that owning a phone makes people live longer?
Back: No. A high correlation does not establish causation; a lurking variable such as a country's wealth likely raises both phone ownership and life expectancy.
Extra: Why students miss this: Observational correlation, however high, cannot by itself prove cause and effect.

Front: Why can a single unusual point have a large effect on the value of r?
Back: Because r is computed from every point's position, one point far from the rest can pull r noticeably up or down — correlation is not resistant to outliers.
Extra: Why students miss this: r is sensitive to outliers; one stray point can make a weak relationship look strong or vice versa.

Front: A dataset has r = 1. What does that say about the scatterplot?
Back: Every point lies exactly on a single straight line with positive slope — a perfect positive linear association.
Extra: Why students miss this: r = 1 does not mean the line is steep or that the slope equals 1; it means the points are perfectly on a line.

Front: Roughly describe what r-values near 0, near ±0.5, and near ±0.9 suggest about linear strength.
Back: Near 0: weak/almost no linear association. Near ±0.5: moderate linear association. Near ±0.9: strong linear association (these are rough guides, always read in context).
Extra: Why students miss this: These cutoffs are informal — there is no exact boundary where a correlation becomes "strong."

Front: What does the least-squares regression line (LSRL) minimize?
Back: The sum of the squared residuals — the squared vertical distances between the observed points and the line.
Extra: Why students miss this: It minimizes squared VERTICAL distances, not perpendicular distances, and squaring is why large misses count heavily.

Q: Fill in the regression equation form: [...], where a is the y-intercept and b is the slope.
A: ŷ = a + bx
Extra: ŷ = a + bx. ŷ is the predicted response, a is the intercept, b is the slope, and x is the explanatory value.
Why students miss this: The left side is ŷ (predicted), not y (an actual observed value).

Front: Why is the output of a regression equation written as ŷ rather than y?
Back: Because the line gives a PREDICTED value of the response for a given x, not the actual observed value — the hat marks it as an estimate.
Extra: Why students miss this: ŷ is rarely equal to any actual y; real points usually fall above or below the line.

Front: A model predicts plant height from days as ŷ = 2.5 + 0.8x (height in cm, x in days). Predict the height at 10 days.
Back: ŷ = 2.5 + 0.8(10) = 2.5 + 8 = 10.5 cm.
Extra: Why students miss this: Plug x into the equation; do not just read the slope or intercept as the answer.

Front: Using ŷ = 30 - 0.5x to predict test anxiety score from hours slept, what score does the model predict for a student who slept 8 hours?
Back: ŷ = 30 - 0.5(8) = 30 - 4 = 26. The model predicts an anxiety score of 26.
Extra: Why students miss this: A predicted value is the model's estimate for that x, not a guaranteed outcome for any individual.

Front: What does the slope b of a regression line describe (descriptively)?
Back: The predicted change in the response variable for each one-unit increase in the explanatory variable.
Extra: Why students miss this: The slope describes the PREDICTED change in ŷ, not an exact change for every individual, and it is descriptive — no inference about a true slope here.

Front: For ŷ = 50 + 12x predicting monthly sales (in $1000s) from ad spend (in $1000s), interpret the slope of 12 in context.
Back: For each additional $1000 spent on ads, the model predicts sales increase by about $12,000 (12 thousand dollars).
Extra: Why students miss this: Always state the slope as a PREDICTED change with the variables' real units and context, not a bare number.

Front: For ŷ = 50 + 12x predicting monthly sales (in $1000s) from ad spend (in $1000s), interpret the y-intercept of 50 in context.
Back: When ad spend is $0, the model predicts sales of about $50,000 (50 thousand dollars).
Extra: Why students miss this: The intercept is the predicted response at x = 0; whether that is meaningful depends on whether x = 0 makes sense.

Front: A model predicts a newborn's weight from gestation days: ŷ = -5 + 0.2x. Why is the y-intercept of -5 not meaningful here?
Back: The intercept is the predicted weight at x = 0 days, but a pregnancy of 0 days is impossible (and a negative weight is nonsense), so the intercept has no real-world meaning.
Extra: Why students miss this: An intercept is only meaningful when x = 0 is a realistic value within the data's context.

Front: A line built from study data covering 1 to 6 hours of practice is ŷ = 40 + 9x. Predicting the score for 20 hours of practice is an example of what, and why is it risky?
Back: It is extrapolation — predicting outside the range of the data (1 to 6 hours). It is risky because the linear pattern may not hold beyond the observed x-values, so the prediction may be far off.
Extra: Why students miss this: A regression line is only trustworthy for x-values within the range of the data used to build it.

Front: Why does extrapolation often produce unreliable predictions?
Back: Because there is no data far outside the observed range to confirm the relationship stays linear there — the true pattern may bend or change, so the line may not apply.
Extra: Why students miss this: A line fitting the data well does not guarantee it works far outside that range.

Front: A line relating a child's age (2 to 10 years) to height predicts a height of over 7 feet at age 40. What mistake does this illustrate?
Back: Extrapolation — using the line far beyond the age range it was built on. Growth slows and stops in adulthood, so the linear trend does not continue.
Extra: Why students miss this: Predictions far outside the data range can be wildly wrong even when the line fit the original data perfectly.

Front: Which point does every least-squares regression line always pass through?
Back: The point of averages (x̄, ȳ) — the mean of x and the mean of y.
Extra: Why students miss this: The line need not pass through any actual data point, but it always passes through (x̄, ȳ).

Front: A model is ŷ = 12 + 3x. An actual data point has x = 4 and observed y = 30. Find the predicted value and the residual.
Back: Predicted ŷ = 12 + 3(4) = 24. Residual = observed - predicted = 30 - 24 = 6.
Extra: Why students miss this: Residual is observed minus predicted (y - ŷ), not predicted minus observed.

Front: Define a residual in regression.
Back: A residual is the observed value minus the predicted value: residual = y - ŷ. It is the vertical distance from a data point to the line.
Extra: Why students miss this: A residual is observed minus predicted, not the other way around; sign matters.

Front: A car's predicted price is ŷ = $18,000 but it actually sold for $16,500. Compute the residual.
Back: Residual = observed - predicted = 16,500 - 18,000 = -$1,500.
Extra: Why students miss this: A negative residual is correct here and means the model overpredicted the price.

Front: What does a positive residual tell you about the model's prediction for that point?
Back: The observed value is greater than the predicted value, so the model UNDERpredicted — the actual point sits above the line.
Extra: Why students miss this: Positive residual = underprediction (point above line); negative = overprediction (point below line).

Front: A weather model predicts 80°F but the day reaches only 74°F. Is the residual positive or negative, and did the model over- or underpredict?
Back: Residual = 74 - 80 = -6, a negative residual. The model overpredicted (the actual value fell below the prediction).
Extra: Why students miss this: A negative residual means the prediction was too high, not too low.

Front: What is the purpose of examining a residual plot?
Back: To check whether a linear model is appropriate — a residual plot shows the residuals against x (or ŷ) so you can look for leftover patterns.
Extra: Why students miss this: A residual plot is not just a fancy scatterplot of the data; it is a tool to judge whether the LINE was the right model.

Front: A residual plot shows points scattered randomly above and below zero with no clear pattern. What does this suggest about the linear model?
Back: A linear model is appropriate — random, patternless scatter around zero means the line captured the relationship well.
Extra: Why students miss this: Random scatter in the residual plot is the SIGN of a good linear fit, not a sign of a problem.

Front: A residual plot shows a clear curved (U-shaped) pattern. What does this tell you?
Back: A linear model is NOT appropriate — the leftover curved pattern means a nonlinear model would fit the data better.
Extra: Why students miss this: Any clear pattern (curve, fanning) in a residual plot signals the line is the wrong model, even if r looks high.

Front: What does the coefficient of determination r² represent?
Back: The proportion (or percentage) of the variation in the response variable that is explained by the linear model with the explanatory variable.
Extra: Why students miss this: r² is a fraction of variation explained, not the fraction of points on the line or the slope.

Front: A regression of weight on height gives r² = 0.64. Interpret this in context.
Back: About 64% of the variation in weight is explained by the linear relationship with height; the remaining 36% is due to other factors.
Extra: Why students miss this: r² = 0.64 does not mean 64% of points lie on the line or that predictions are 64% accurate.

Front: What is the key difference between r and r²?
Back: r measures the direction and strength of the linear association (signed, from -1 to 1). r² is the proportion of variation in the response explained by the model (always 0 to 1, no sign).
Extra: Why students miss this: r² loses the sign, so it cannot tell you direction — only r reveals whether the association is positive or negative.

Front: A regression has r = -0.7. Find r² and state what information is lost when you square r.
Back: r² = (-0.7)² = 0.49. Squaring discards the sign, so r² (0.49) no longer tells you the association is negative.
Extra: Why students miss this: You cannot recover the direction from r² alone; r² = 0.49 could come from r = 0.7 or r = -0.7.

Front: Why is r² described as the "proportion of variation explained"?
Back: Because using the regression line (instead of just the mean of y) reduces the leftover variation in the response; r² is the fraction of the total variation in y that the line accounts for.
Extra: Why students miss this: The unexplained variation (1 - r²) comes from other factors and natural scatter, not from a flaw in arithmetic.

Front: Two students debate a model with r = 0.6. One says "60% of the variation is explained." Is that right? Show the correct value.
Back: No. The explained variation is r² = 0.6² = 0.36, so only about 36% of the variation in the response is explained — not 60%.
Extra: Why students miss this: Never report r itself as the percentage of variation explained; you must square it first.

Front: What does the standard deviation of the residuals (often called s) tell you about a regression model?
Back: It is the typical size of a prediction error — roughly how far, on average, the observed values fall from the line (in the response variable's units).
Extra: Why students miss this: Smaller s means the points cluster tightly around the line (better predictions); it is measured in the units of y, not unitless like r.

Front: In regression, what makes a point an outlier?
Back: A point with a large residual — it lies far from the regression line in the y-direction (its observed value differs greatly from the prediction).
Extra: Why students miss this: A regression outlier is defined by a large residual (far from the line), not simply by an extreme x-value.

Front: What is a high-leverage point?
Back: A point with an x-value far from the mean of the other x-values — it sits at the extreme left or right of the scatterplot in the explanatory direction.
Extra: Why students miss this: Leverage is about an extreme x-value, which is different from having a large residual.

Front: What makes a point influential in a regression?
Back: A point is influential if removing it would noticeably change the regression line (its slope or intercept), the correlation, or r². High-leverage points are often influential.
Extra: Why students miss this: Influence is judged by the EFFECT of removing the point, not just by whether it looks unusual.

Front: A scatterplot has one point at a far-right x-value sitting well off the trend. After it is removed, the slope changes from 0.3 to 0.9. What kind of point was it?
Back: An influential point (and high-leverage, since its x-value is extreme) — removing it substantially changed the slope of the regression line.
Extra: Why students miss this: A point can be influential even if it does not have the single largest residual, especially when its x-value is extreme.

Front: Why can a high-leverage point be more influential on the regression line than a point with a large residual near the center?
Back: Because an extreme x-value gives a point more pull on the line's slope — like sitting far out on a seesaw — so it can swing the line more than a point near the average x, even one with a sizable residual.
Extra: Why students miss this: A large residual alone does not make a point influential; its position in x (leverage) strongly affects how much it moves the line.

Front: A teacher adds one student who studied far more hours than anyone and also scored far higher; r jumps from 0.45 to 0.80. How should you describe this new point's effect?
Back: It is an influential, high-leverage point: its extreme x-value pulled the correlation and the line, making the linear association look much stronger than the rest of the data shows.
Extra: Why students miss this: A single influential point can inflate r and the slope, so a strong-looking relationship may rest on one unusual observation.