📊 AP Statistics FRQ Study Plan

"A researcher collects a random sample of 247 adults and records their scores on a 100-point scale. Describe the shape, center, and spread of the distribution. If the mean is 68.3 with a standard deviation of 12.1, what proportion of scores fall between 56.2 and 80.4? Explain your reasoning."

"symmetric/unimodal/bimodal/skewed right/skewed left" · "median is more appropriate than mean for skewed distributions" · "outliers identified using 1.5×IQR rule" · "approximately normal distributions follow the empirical rule (68-95-99.7)" · "z-scores measure standard deviations from the mean"

"A city planner collects data on the number of bus stops (x) and average daily ridership (y) for 18 city zones. Create a scatterplot, describe the relationship, calculate and interpret the correlation coefficient r, and determine whether a linear model is appropriate. If the regression equation is ŷ = 142.3 + 8.7x, interpret the slope and predict ridership for a zone with 25 bus stops."

"positive/negative association" · "strength: strong/moderate/weak" · "linearity — look at residual plot" · "r² = [value] means [X] explains [Y]% of the variability in [Y]" · "lurking variables" · "correlation does not imply causation" · "ŷ = a + bx: for each increase of 1 in x, y increases by b units"

"A study wants to determine whether a new tutoring program improves math test scores. Describe how you would design an experiment to answer this question. Explain how you would address confounding variables, randomization, and control groups. Suppose 200 students are randomly assigned to treatment or control. What are the experimental units? What is the response variable? What is a potential placebo effect?"

"randomized comparative experiment" · "completely randomized design" · "control group" · "treatment group" · "double-blind" (if evaluator doesn't know group) · "random assignment" vs "random sampling" — know the difference and when each applies · "statistical significance requires random assignment" · "generalizability requires random sampling"

"A game at a carnival pays $5 to play. A player draws two cards from a standard deck without replacement. If both cards are hearts, the player wins $20. Otherwise, the player wins nothing. Define X as the player's net gain. Construct the probability distribution for X. Find the expected value E(X) and standard deviation σₓ. Based on your calculations, would you recommend playing this game? Justify your answer."

"P(X = k) = ..." · "expected value E(X) = Σx·P(X=x)" · "E(X) = np for binomial" · "σ²(X) = Σ(x−μ)²·P(X=x)" · "standard deviation tells us typical deviation from mean" · "linear transformation: E(aX+b) = aE(X)+b" · "if E(X) < 0, the player loses money on average"

"A manufacturer claims that 15% of the batteries they produce are defective. A quality inspector randomly samples 50 batteries. Describe the sampling distribution of p̂ (the sample proportion of defectives). Calculate the probability that p̂ exceeds 0.20. Then, the inspector samples 200 batteries instead. How does this change the sampling distribution? Explain why the Central Limit Theorem applies here."

"CLT: distribution of p̂ is approximately normal with mean μₚ̂ = p₀ and SE = √(p₀(1−p₀)/n)" · "conditions: random sample, independence (n < 10% rule), large enough sample (np ≥ 10 AND n(1−p) ≥ 10)" · "standard error vs standard deviation" · "as n increases, SE decreases (estimates get more precise)"

"A sociologist wants to estimate the average hours per week that high school seniors spend on homework. A random sample of 64 seniors gives a sample mean of 5.2 hours and a sample standard deviation of 2.8 hours. Construct and interpret a 95% confidence interval for the true mean μ. State and verify the conditions for inference. What does '95% confidence' actually mean in this context?"

"conditions: random sample ( SRS or random assignment ), independence (10% condition), nearly normal (n≥30 or graph shows no strong skew/outliers)" · "x̄ ± t*·(s/√n)" · "t* depends on df = n−1 and confidence level" · "interpretation: 'We are 95% confident the true mean is between [a] and [b]'" · "confidence is about the method, not any one interval"

"A cereal company claims that the mean weight of their '20-ounce' boxes is at least 20 ounces. A consumer group believes the true mean is less than 20 ounces. They randomly select 40 boxes and find x̄ = 19.85 oz with s = 0.42 oz. Test the company's claim at α = 0.05. State the null and alternative hypotheses, check conditions, calculate the test statistic, find the p-value, and state your conclusion in context."

"H₀: μ = μ₀ (null = no effect / status quo)" · "Hₐ: μ < μ₀ (or > μ₀ or ≠ μ₀ — match the scenario)" · "t = (x̄ − μ₀) / (s/√n)" · "p-value = P(t ≥ observed | H₀ is true)" · "if p < α → reject H₀, evidence supports Hₐ" · "Type I error: reject H₀ when it's actually true" · "Type II error: fail to reject H₀ when Hₐ is actually true"

"A fitness coach hypothesizes that a new 8-week training program will lower resting heart rate. Twenty athletes are randomly assigned to the program (treatment) or a control group. After 8 weeks, the treatment group's mean heart rate is 62.3 bpm (sd = 4.1) and the control group's mean is 65.8 bpm (sd = 3.9). Construct a 95% confidence interval for the difference in mean heart rates. Does this support the coach's hypothesis? Conduct an appropriate significance test."

"paired vs two-sample: was the same subject measured twice? → use paired t-test" · "two-sample t: (x̄₁ − x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂)" · "Welch's t (unequal variances) is preferred when s₁ ≠ s₂" · "df = Welch-Satterthwaite approximation" · "if CI for μ₁ − μ₂ does not include 0 → statistically significant difference"

"In a random sample of 850 residents, 306 reported that they would support a new public transit measure. Construct a 95% confidence interval for the true proportion of residents who support the measure. The transit authority claims at least 38% support. Test the authority's claim at the 5% significance level. Based on your analysis, should the transit authority proceed with the measure?"

"one-proportion z interval: p̂ ± z*·√(p̂(1−p̂)/n)" · "conditions: random sample, independence (n < 10% population), normal (np̂ ≥ 10 and n(1−p̂) ≥ 10)" · "one-proportion z test: H₀: p = p₀" · "z = (p̂ − p₀) / √(p₀(1−p₀)/n)" · "for confidence interval: use p̂ in SE" · "for hypothesis test: use p₀ in SE (null proportion)"

"A market researcher asks whether brand preference (A, B, C, or D) is independent of age group (18–29, 30–49, 50+). Survey 400 consumers and record responses in a 3×4 table. Perform a chi-square test of independence. State H₀ and Hₐ, check conditions, calculate the test statistic and p-value, and state your conclusion at α = 0.05. If significant, which cell(s) contribute most to the test statistic?"

"H₀: the two variables are independent" · "Hₐ: the two variables are associated" · "χ² = Σ(O−E)²/E where Eᵢⱼ = (row total × col total) / grand total" · "conditions: random sample, independence (expected count ≥ 5 in ALL cells — not observed, expected!)" · "df = (rows−1)(cols−1)" · "chi-square is ALWAYS one-tailed (right tail)" · "larger χ² → stronger evidence against H₀"

"A biologist collects data on height (x, in cm) and wingspan (y, in cm) for 22 birds and fits a least-squares regression line: ŷ = 4.2 + 0.87x with sₑ = 3.1 and R² = 0.83. Test whether there is a significant linear relationship between height and wingspan at the 5% level. Provide a 95% confidence interval for the slope parameter β. Interpret both the test and the interval in context. If a bird is 40 cm tall, predict its wingspan and give a prediction interval."

"H₀: β = 0 (no linear relationship)" · "Hₐ: β ≠ 0 (linear relationship exists)" · "t = b / SE_b where SE_b = sₑ / √Σ(xᵢ − x̄)²" · "df = n − 2" · "b ± t*·SE_b for CI for slope" · "R² = [value] means [X] explains [Y]% of variation in [Y]" · "prediction interval vs confidence interval — prediction is WIDER because it includes individual variation"

"A researcher compares the effectiveness of four different study methods on exam scores. Thirty-two students are randomly assigned to one of four methods (8 per group). The mean scores are: Method A = 78, Method B = 82, Method C = 75, Method D = 85, with MSbetween = 312 and MSwithin = 148. Conduct a one-way ANOVA at α = 0.05. State H₀ and Hₐ, calculate the F-statistic, state degrees of freedom, and conclude. If significant, which groups differ? Explain how you would follow up."

"H₀: μ₁ = μ₂ = μ₃ = μ₄ (all group means equal)" · "Hₐ: at least one μᵢ is different" · "F = MSbetween / MSwithin" · "df between = k−1, df within = N−k" · "conditions: independent SRS from each group, independent groups, normally distributed populations (or large n), equal standard deviations (or check with boxplots)" · "if significant → use Tukey or Bonferroni for follow-up pairwise comparisons"