Practice Test 4 - AP Statistics

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

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1. What is the probability that a randomly selected customer will purchase an extended warranty?
2. Two sections of the same course have exam scores that are approximately Normal with the same mean $\mu=75$. Section 1 has $\sigma=8$ and Section 2 has $\sigma=12$. A score is considered "high" if it is at least 90. Without computing any exact probabilities, which comparison is supported?
3. An economist studies whether higher interest rates are associated with fewer new home loans. For a random sample of 36 months, the economist records the average interest rate (%) ($x$) and the number of new home loans issued ($y$), then fits a regression model predicting loans from interest rate. The research claim is that as interest rates rise, the predicted number of loans falls (negative slope). Which hypotheses are appropriate for testing this claim about the population regression slope?
4. A teacher investigates whether the number of practice problems completed ($x$) predicts score on a unit test ($y$) for students in a large district. A random sample of 40 students is selected, and a regression line is fit to predict test score from number of practice problems. The research claim is that completing more practice problems increases the predicted test score (positive slope). Which hypotheses are appropriate for testing this claim about the population regression slope?
5. A city planner models commute time (minutes) as a linear function of distance from downtown (miles) using a random sample of commuters. A 90% confidence interval for the slope is $(0.6,\ 1.4)$. Which interpretation is correct?
6. A coffee shop tests whether two types of music (jazz vs. pop) affect average customer time spent in the shop. The manager knows Saturdays are much busier than Tuesdays and expects that day-of-week strongly influences time spent. The shop can play only one music type per day, and the study will run for 8 days total (4 Tuesdays and 4 Saturdays over a month). Which experimental design is most appropriate to compare the music types while accounting for day-of-week?
7. A hospital compares two methods for reducing patient wait times in an emergency department. Over the next month, on randomly selected days the hospital uses Method A, and on the remaining days it uses Method B. For each day, the hospital records the mean patient wait time (in minutes) for that day. Which inference procedure is most appropriate to test whether the mean daily wait time differs between the two methods?
8. A city health department claims that fewer than 30% of adults in the city have received a flu shot this season. To test this claim, a random sample of 250 adults is surveyed, and 62 report having received a flu shot (sample proportion $\hat{p}=0.248$). Which hypotheses are appropriate for a one-proportion $z$ test of the department's claim?
9. A school district tests whether a new online tutoring program increases the mean SAT math score compared with the historical mean of 520. The hypotheses are $H_0: \mu = 520$ versus $H_a: \mu > 520$, using $\alpha = 0.05$. After collecting a random sample of students who used the program, the district performs the test and fails to reject $H_0$. In reality, the tutoring program truly does increase the mean score above 520 for the population of participating students. Which type of error was made?
10. A materials engineer tests whether average tensile strength differs between a new alloy and a current alloy. A random sample of 32 specimens of the new alloy had a mean tensile strength of 512 MPa, and a random sample of 30 specimens of the current alloy had a mean of 505 MPa. Let $\mu_{new}$ and $\mu_{cur}$ be the true mean tensile strengths for the new and current alloys. Which hypotheses are appropriate?
11. A coffee shop owner randomly samples 18 days and records the mean number of customers per day. Using a one-sample $t$ procedure, the owner reports a 90% confidence interval for the population mean number of customers per day of $(118,\ 134)$. Which interpretation is correct?
12. A public health researcher randomly sampled 600 adults in a city to estimate the proportion $p$ of all adults in the city who got a flu shot this year. In the sample, 318 reported getting a flu shot. A 99% confidence interval for $p$ is $(0.49,\ 0.57)$. Which interpretation is correct?
13. A school district randomly sampled 240 high school students and found that 156 reported getting at least 8 hours of sleep on a typical school night. A 95% confidence interval for the true proportion $p$ of all high school students in the district who get at least 8 hours of sleep is $(0.59,\ 0.71)$. Which interpretation is correct?
14. A doctor studies whether the number of minutes of exercise per day ($x$) predicts fasting blood glucose in mg/dL ($y$). The doctor expects that more exercise is associated with lower fasting blood glucose in the population. Which hypotheses are appropriate for testing this expectation about the population regression slope?
15. A school district studies whether the number of hours a student spends on a new online math practice platform each week ($x$) helps predict the student's end-of-term math exam score ($y$). Using data from 60 randomly selected students, the district fits a least-squares regression of exam score on practice hours and wants to test the research claim that more practice is associated with higher average exam scores (a positive population slope). Which hypotheses are appropriate for a significance test of the regression slope?
16. A city planner records data from 15 neighborhoods on $x$ = distance (miles) from downtown and $y$ = average monthly rent (dollars). A least-squares regression of rent on distance gives a 90% confidence interval for the slope $\beta$ of $(-220,\ -40)$. Which interpretation is correct?
17. A nutritionist tests whether the mean sodium content of a brand of soup is less than the label claim of 800 mg. The hypotheses are $H_0: \mu=800$ versus $H_a: \mu<800$, at $\alpha=0.05$. The p-value is 0.60. Which interpretation of the p-value is correct?
18. A hospital records the waiting time (in minutes) in an emergency room. The population distribution is strongly right-skewed due to occasional very long waits. The hospital administrator takes many random samples of $n=40$ patients and calculates the sample mean waiting time $\bar{x}$ for each sample, forming the sampling distribution of $\bar{x}$. Why is the sampling distribution approximately normal?
19. A company wants to estimate the proportion of all employees nationwide who are satisfied with their health benefits. The company divides employees into regions (Northeast, South, Midwest, West) and randomly selects 50 employees from each region to survey, for a total of 200 employees. Which statement best describes the sample representativeness?
20. A museum estimates that 27% of visitors buy a souvenir. Many random samples of $n=90$ visitors are taken and $\hat p$ is computed each time. Which statement about the sampling distribution is correct?

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