Format: Section I — 20 multiple-choice questions · Section II — 2 free-response questions
Suggested time: 40 min (MCQ) + 30 min (FRQ) · 29 points in total · calculator permitted throughout
Coverage: Exploring One-Variable Data · Exploring Two-Variable Data · Collecting Data · Probability, Random Variables and Probability Distributions · Sampling Distributions · Inference for Proportions and Means.
Standard normal values you may need: P(Z < 1) ≈ 0.841, P(Z < 1.96) ≈ 0.975, P(Z < 2) ≈ 0.977.
Section I — Multiple Choice (20 points, 1 point each)
1. The median is generally preferred to the mean as a measure of center when the data are
A) skewed or contain outliers B) symmetric C) categorical D) approximately normal
2. A test score has a z-score of −1.5. This means the score is
A) at the 15th percentile B) 1.5 points below the mean C) impossible, because scores cannot be negative D) 1.5 standard deviations below the mean
3. In a strongly right-skewed distribution, the mean is typically
A) equal to the median B) less than the median C) greater than the median D) equal to the mode
4. A data set has Q1 = 12 and Q3 = 20. Using the 1.5 × IQR rule, which of these values is an outlier?
A) 2 B) 34 C) 30 D) 31
5. A data set has standard deviation 2. Each value is multiplied by 3 and then 4 is added. The new standard deviation is
A) 10 B) 2 C) 18 D) 6
6. A simple random sample of size n is one in which
A) people are chosen because they are easy to reach B) every possible group of n individuals has the same chance of being selected C) only volunteers take part D) the first n people on a list are taken
7. Which of these is most likely to produce a biased estimate?
A) an online poll that anyone can choose to answer B) random assignment of treatments C) blinding the subjects D) a larger random sample
8. Exam scores are approximately normal with mean 70 and standard deviation 5. About what percentage of scores lie between 60 and 80?
A) 68% B) 99.7% C) 95% D) 47.5%
9. In an experiment, the main purpose of a control group is to
A) increase the sample size B) introduce variability C) avoid the need for random assignment D) provide a baseline for comparison
10. Events A and B are independent, with P(A) = 0.3 and P(B) = 0.5. P(A and B) =
A) 0.8 B) 0.2 C) 0.15 D) 0.65
11. A residual in a regression is calculated as
A) observed y − predicted y B) predicted y − observed y C) the slope of the line D) the correlation coefficient
12. A correlation of r = −0.9 between two variables indicates
A) a weak negative linear relationship B) a strong negative linear relationship C) no relationship D) that one variable causes the other
13. Increasing the sample size from 100 to 400 (with everything else unchanged) makes the standard error of the sample mean
A) four times as large B) half as large C) one quarter as large D) unchanged
14. A 95% confidence interval for a population mean is (48, 56). Which statement is correct?
A) 95% of the data lie between 48 and 56. B) There is a 95% chance that the population mean changes between samples. C) 95% of sample means lie between 48 and 56. D) We are 95% confident that the interval from 48 to 56 captures the true population mean.
15. A researcher divides a school into grade levels and takes a random sample of students from each grade. This is
A) stratified random sampling B) cluster sampling C) convenience sampling D) a census
16. Adult heights are approximately normal with mean 170 cm and standard deviation 8 cm. The proportion of adults taller than 186 cm is closest to
A) 0.05 B) 0.16 C) 0.023 D) 0.977
17. A fair coin is tossed 4 times. The probability of exactly 2 heads is
A) 0.25 B) 0.375 C) 0.5 D) 0.0625
18. A random variable X takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3. The expected value of X is
A) 1.0 B) 1.2 C) 0.9 D) 1.1
19. In a hypothesis test, a Type I error occurs when we
A) reject H₀ when H₀ is true B) fail to reject H₀ when H₀ is false C) reject H₀ when H₀ is false D) fail to reject H₀ when H₀ is true
20. In a random sample of 100 voters, the true proportion who support a measure is p = 0.4. The standard deviation of the sample proportion is about
A) 0.024 B) 0.0024 C) 0.049 D) 0.49
Section II — Free Response (9 points)
FRQ 1 (Regression — 5 points).
A teacher records the hours studied (x) and exam score (y) for 8 students. The least-squares regression line is ŷ = 40 + 8x, and the correlation is r = 0.85.
(a) Interpret the slope of the regression line in context. (1 pt)
(b) Predict the exam score of a student who studies for 5 hours. (1 pt)
(c) A student who studied for 5 hours scored 75. Calculate and interpret the residual for this student. (1 pt)
(d) Calculate r² and interpret it in context. (1 pt)
(e) Does this study show that studying more causes higher exam scores? Explain. (1 pt)
FRQ 2 (Inference for a Proportion — 4 points).
In a random sample of 200 students at a large high school, 124 said they usually sleep for less than 8 hours on school nights.
(a) Identify the parameter of interest and calculate the sample proportion. (1 pt)
(b) Check the conditions for constructing a confidence interval for a population proportion. (1 pt)
(c) Construct a 95% confidence interval for the population proportion. (1 pt)
(d) The principal claims that "most students at this school sleep for less than 8 hours on school nights." Does your interval support this claim? Explain. (1 pt)
Answer key (Section I)
| Q | Ans | Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|---|---|
| 1 | A | 6 | B | 11 | A | 16 | C |
| 2 | D | 7 | A | 12 | B | 17 | B |
| 3 | C | 8 | C | 13 | B | 18 | D |
| 4 | B | 9 | D | 14 | D | 19 | A |
| 5 | D | 10 | C | 15 | A | 20 | C |
Key distribution: A×5, B×5, C×5, D×5.
Worked solutions (Section I)
1. (A) The median is resistant to extreme values; the mean is pulled toward the tail or outliers. For symmetric data the two are similar.
2. (D) A z-score counts standard deviations from the mean; the negative sign means below the mean. (1.5 points confuses standard deviations with raw units.)
3. (C) The long right tail pulls the mean upward, above the median.
4. (B) IQR = 20 − 12 = 8, and 1.5 × 8 = 12. The fences are 12 − 12 = 0 and 20 + 12 = 32, so only 34 lies outside them.
5. (D) Multiplying by 3 multiplies the spread by 3; adding 4 shifts every value and does not change the spread. New SD = 3 × 2 = 6. (10 wrongly adds the 4.)
6. (B) This is the definition of an SRS. The other options describe convenience sampling, voluntary response and systematic selection without randomness.
7. (A) Voluntary response over-represents people with strong opinions. The other three are good practice and do not cause bias.
8. (C) 60 and 80 are 2 standard deviations either side of the mean, and the empirical rule puts about 95% of a normal distribution within 2 SD.
9. (D) The control group shows what happens without the treatment, so the treatment effect can be measured against it.
10. (C) For independent events, P(A and B) = P(A) × P(B) = 0.3 × 0.5 = 0.15. (0.8 adds the probabilities.)
11. (A) Residual = observed − predicted. A positive residual means the point lies above the line.
12. (B) |r| = 0.9 is close to 1, so the linear relationship is strong; the sign shows it is negative. Correlation alone never shows causation.
13. (B) SE = σ/√n. Multiplying n by 4 divides the standard error by √4 = 2. (One quarter forgets the square root.)
14. (D) The confidence level describes the method: 95% of intervals built this way capture the true parameter. It says nothing about where individual data values lie.
15. (A) Sampling randomly within every group is stratified sampling. Cluster sampling would select whole groups at random and survey everyone in them.
16. (C) z = (186 − 170)/8 = 2, and P(Z > 2) ≈ 0.023. (0.977 is P(Z < 2); 0.05 counts both tails.)
17. (B) Binomial: ⁴C₂ × 0.5² × 0.5² = 6/16 = 0.375.
18. (D) E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6 = 1.1.
19. (A) A Type I error is a false alarm: rejecting a true null hypothesis. Failing to reject a false null is a Type II error; the other two are correct decisions.
20. (C) √(p(1 − p)/n) = √(0.4 × 0.6/100) = √0.0024 ≈ 0.049. (0.0024 is the variance, before the square root.)
FRQ solutions
FRQ 1 (5 pts).
(a) (1) For each additional hour studied, the predicted exam score increases by 8 points.
(b) (1) ŷ = 40 + 8(5) = 80 points.
(c) (1) Residual = observed − predicted = 75 − 80 = −5. The student scored 5 points lower than the line predicts.
(d) (1) r² = 0.85² ≈ 0.72. About 72% of the variation in exam scores is explained by the linear relationship with hours studied.
(e) (1) No. This is an observational study, not an experiment, so a lurking variable (for example, prior ability or motivation) could explain the association. Only random assignment of study time would allow a cause-and-effect conclusion.
FRQ 2 (4 pts).
(a) (1) The parameter p is the proportion of all students at the school who usually sleep less than 8 hours on school nights. p̂ = 124/200 = 0.62.
(b) (1) Random: the sample is random. 10% condition: 200 is less than 10% of the students at a large school. Large counts: np̂ = 124 and n(1 − p̂) = 76 are both at least 10.
(c) (1) 0.62 ± 1.96 × √(0.62 × 0.38/200) = 0.62 ± 1.96 × 0.0343 = 0.62 ± 0.067, giving (0.553, 0.687).
(d) (1) Yes. Every value in the interval is above 0.5, so the interval gives convincing evidence that more than half of the students sleep less than 8 hours on school nights.
Scoring guidance
- Section I: 20 points (1 each).
- Section II: FRQ 1 = 5 points, FRQ 2 = 4 points.
Pedagogy: AP Statistics answers must be in context. "The residual is −5" is incomplete; "the student scored 5 points lower than predicted" earns the point. Name the variables every time.