Practice Test 43 - AP Statistics

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A school nurse expects that about 70% of students will choose the flu shot when it is offered for free. In a particular week, the nurse offered the shot to a random sample of 20 students and observed that only 8 chose to get it (40%). The nurse expected the sample proportion to be near 70%, but the observed result was much lower. Is the result unexpected, given the natural variability in random sampling from a population where the true proportion might be around 70%?
2. A political analyst claims that exactly 50% of adults in a state approve of the governors performance. A random sample of 1000 adults finds that 540 approve (so $ \hat{p}=540/1000$). Which hypotheses are appropriate to test whether approval differs from the analysts claim?
3. A music streaming service tracks whether a user listens to a curated playlist (event $C$) and whether the user upgrades to a paid plan that day (event $U$). The service suspects that curated playlists encourage upgrades. How does knowing that a user listened to a curated playlist ($C$ occurred) affect the likelihood that the user upgraded ($U$)?
4. A manufacturer tests whether the proportion of defective parts exceeds 2%. The hypotheses are $H_0: p = 0.02$ and $H_a: p > 0.02$. After sampling, the company fails to reject $H_0$ at $\alpha = 0.05$. In reality, the true defect rate is $p = 0.02$. Which type of error was made?
5. A car manufacturer claims the mean fuel efficiency for a model is $\mu=35$ mpg. A random sample of 20 cars of this model is tested, and a one-sample $t$ test is performed: $H_0:\mu=35$ vs. $H_a:\mu<35$, using $\alpha=0.05$. The p-value is 0.004. What conclusion is appropriate?
6. A botanist plants 18 seeds in identical pots and records whether each seed germinates. A success is "germinates," and exactly 18 seeds are planted under the same conditions. Does this meet binomial conditions for the number of successes?
7. A biologist creates a scatterplot of the height versus the weight of a sample of adult elephants. The points on the plot appear to follow a pattern that is reasonably straight. How would the form of this relationship be described?
8. A factory inspects 50 light bulbs from a production line. A bulb is a success if it is defective (the factory is tracking defects), and the long-run defect rate is $0.04$ per bulb (assume independence). Which values of $n$ and $p$ correctly model the number of successes with a binomial distribution?
9. A public transit agency samples commute times on a bus route to decide whether to add an express bus during rush hour. If the sample suggests the mean commute time exceeds a target, the agency will reallocate drivers and buses to add service. If the inference is wrong, riders could continue to face long delays or resources could be diverted from other routes that need them more. Why is it important to consider potential error when using the sample to make this operational decision?
10. A national poll claims that 45% of adults have a favorable opinion of a certain policy. A local community leader believes the proportion in their town is different. A random sample of 200 town residents is surveyed. Which statement correctly identifies the inference procedure and null hypothesis?
11. What is the primary purpose of constructing a scatterplot in a statistical analysis involving two variables?
12. A city's transportation department believes that 65% of commuters prefer public transit over driving. A random sample of 30 commuters is surveyed at a transit hub; 14 say they prefer public transit (46.7%). The department asks whether this difference could reasonably be due to random sampling variability with $n=30$ if the true preference rate is 65%. Is the result unexpected?
13. A fitness coach compares the number of push-ups completed in 1 minute by two groups: athletes who warmed up (Warm-up) and athletes who did not (No warm-up). The side-by-side boxplot summaries below include an outlier note. Which comparison is supported? | Feature | Warm-up | No warm-up | |---|---| | Median | about 42 | about 36 | | IQR | about 10 | about 14 | | Outliers | none noted | one high outlier around 70 |
14. A gym tracked whether members attend group classes (Yes/No) and whether they renewed their membership (Renewed/Not). The goal is to compare renewal rates between class attendees and nonattendees using conditional proportions. Which comparison is supported by the data? Counts: Classes: Renewed 96, Not 24 (Total 120). No classes: Renewed 140, Not 60 (Total 200).
15. What is the expected number of red marbles and the standard deviation of the number of red marbles?
16. A jar contains 5 red, 3 blue, and 2 green marbles. One marble is selected at random. Let Event A be "the marble is red" and Event B be "the marble is blue." Which statement about Events A and B is correct?
17. A wildlife biologist compares the proportion of tagged fish recaptured in two lakes using independent random samples of tagged fish. In Lake A, 52 of 160 tagged fish were recaptured; in Lake B, 60 of 150 tagged fish were recaptured. A 90% confidence interval for $p_A - p_B$ is $(-0.17,\ -0.01)$. Which interpretation is correct?
18. A fitness researcher wants to test whether a new 4-week interval-training program improves 1-mile run time compared with the school's usual steady-jog program. Sixty volunteer students are enrolled; each student's baseline mile time is recorded, then students are randomly assigned to follow either the interval plan or the steady-jog plan for 4 weeks. At the end, each student runs a timed mile again, and the change in time (post minus pre) is recorded. Which feature of this design allows the researcher to make a cause-and-effect conclusion about the training program's effect on mile-time improvement?
19. A cereal company states that 40% of its boxes contain a prize. A student repeatedly takes many random samples of $n=100$ boxes from this process and records the sample proportion $\hat p$ with a prize each time. Which statement about the sampling distribution is correct?
20. A company tests whether three different training programs lead to different distributions of certification outcomes. They randomly assign 60 new employees to Program A, 55 to Program B, and 65 to Program C. After training, each employee's outcome is recorded as Pass, Fail, or No-show. Results are summarized below. Which chi-square test is appropriate, and what are the correct hypotheses? (There are multiple groups due to random assignment to programs.)

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