Practice Test 74 - AP Statistics

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

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1. In an AP Statistics class, let $X$ be the number of minutes a student spends on math homework in an evening with mean $\mu_X=50$ and standard deviation $\sigma_X=10$. Let $Y$ be the number of minutes the same student spends on reading with mean $\mu_Y=30$ and standard deviation $\sigma_Y=6$. Assume $X$ and $Y$ are independent. Define the combined random variable $T=X+Y$ (total homework time). Which statement about $T$ is correct?
2. A call center knows that the true proportion of calls resolved on the first call is $p=0.64$. A manager repeatedly takes many random samples of size $n=16$ from all calls and computes $\hat p$ each time. Which statement about the sampling distribution of $\hat p$ is correct?
3. A bank uses a model (built from a sample of past borrowers) to predict which applicants are likely to default. Based on the model, the bank plans to automatically deny loans to applicants above a risk threshold. Denying too many loans could unfairly exclude qualified borrowers; approving too many could increase losses and raise interest rates. Why is it important to consider potential error?
4. A company wants to estimate the proportion of its customers who prefer email notifications over text notifications. Two independent simple random samples of 80 customers are selected from the same customer list. In Sample 1, 41 customers prefer email; in Sample 2, 50 customers prefer email. The sampling frame and question were the same for both samples. Why might the sample results differ?
5. A company compares the proportion of defective items from Machine A and Machine B. A 90% confidence interval for the difference in proportions, defined as $p_A - p_B$, is $(0.01,\ 0.06)$. An engineer claims, "Machine A produces a higher proportion of defective items than Machine B." Is the claim supported by the confidence interval?
6. A real estate analyst regresses house sale price (in thousands of dollars) on house size (in square feet) using a random sample of 50 recent sales in a city. A 95% confidence interval for the slope is $(0.08,\ 0.15)$ thousand dollars per square foot. Which interpretation is correct?
7. A librarian recorded the number of books checked out by 60 patrons in one week. The five-number summary (books) is: min $=0$, $Q_1=1$, median $=2$, $Q_3=4$, max $=15$. A standard boxplot is drawn. Which feature is consistent with the summary statistics?
8. A researcher calculates a sample regression line for the relationship between caffeine intake and hours of sleep for a random sample of students. She knows that if she took another random sample, she would likely get a different sample slope. How does this concept of sampling variability in the slope affect her conclusions about the true relationship?
9. A student surveys 30 households and records whether each household owns a dog. A success is "owns a dog," and exactly 30 households are surveyed, but the student chooses only houses on the same street, where dog ownership is unusually high. Does this meet binomial conditions to model the number of successes?
10. A nonprofit claims that more than 55% of donors prefer to give online rather than by mail. A random sample of 220 donors is selected, and 112 prefer online giving ($\hat{p}=0.509$). Which hypotheses are appropriate for testing the nonprofit's claim about the population proportion $p$?
11. A city water department recorded the daily water use (in thousands of gallons) for a random sample of 40 households in the city. The five-number summary was: min $=1.2$, $Q_1=2.0$, median $=2.6$, $Q_3=3.4$, max $=6.1$. A boxplot is drawn using these values (no outliers are marked separately). Which feature is consistent with the summary statistics?
12. An online retailer tracks whether a customer clicks on a product review section (event $R$) and whether the customer completes a purchase (event $P$). The retailer believes review readers are more serious shoppers. How does knowing that a customer clicked on reviews ($R$ occurred) affect the likelihood that the customer completed a purchase ($P$)?
13. A physical therapist wants to compare two stretching routines (Routine 1 vs Routine 2) on hamstring flexibility. Each participant has two legs, and flexibility can differ between a person's left and right leg. The therapist can assign different routines to different legs of the same person and measure flexibility after 4 weeks. Which experimental design is most appropriate?
14. A student says they generated a random sequence of 30 outcomes from a fair coin, but the sequence contains no runs longer than 2 (no HHH or TTT anywhere). The total is 16 heads and 14 tails. Is the pattern consistent with random behavior? Under randomness, you generally expect to see some runs of length 3 or more in 30 flips; having none may be suspiciously "pattern-avoiding."
15. A nutritionist claims that the proportion of clients who meet a daily fiber goal is higher for clients using a meal-planning app than for clients not using the app. In a sample, 68 of 120 app users met the goal, and 52 of 125 non-users met the goal. Which hypotheses are appropriate?
16. At a school carnival, a student randomly selects one ticket from a box containing 40 tickets: 18 are blue, 12 are red, and 10 are green. Some tickets are marked as "prize" tickets: 14 of the 40 tickets are prize tickets, including 6 blue prize tickets, 5 red prize tickets, and 3 green prize tickets. Let Event A be "the ticket is blue" and Event B be "the ticket is a prize ticket." Which statement about Events A and B is correct?
17. A school principal wants to determine whether allowing students to listen to music during independent work improves quiz scores. Two existing classes of the same course taught by different teachers were used: one class was allowed to listen to music during work time for three weeks, and the other class was not. At the end, the music class had a higher average quiz score. Which conclusion is justified based on the design of this study?
18. A school nurse wants to estimate the proportion of students who get at least 8 hours of sleep on school nights. The nurse plans to survey only students who come to the nurse's office during one week. The research question is: "What proportion of all students at this school get at least 8 hours of sleep on school nights?" Which aspect is most important to address before collecting data?
19. A gym records the number of minutes 60 adult members spend exercising during a single visit. The five-number summary (in minutes) is: min $=12$, $Q_1=28$, median $=41$, $Q_3=55$, max $=96$. A boxplot is drawn using these values (no outliers marked). Which feature is consistent with the summary statistics?
20. A city transportation office claims that the proportion of commuters who are satisfied with bus service is higher among riders on routes with new buses than among riders on routes with older buses. In a sample, 92 of 140 riders on new-bus routes reported being satisfied, and 80 of 150 riders on old-bus routes reported being satisfied. Which hypotheses are appropriate?

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