Practice Test 14 - AP Statistics

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

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1. In a large city, the distribution of adult male heights is approximately Normal with mean $\mu=70$ inches and standard deviation $\sigma=3$ inches for the population of adult males. The marked value is $x=76$ inches. Which statement about the marked value is correct?
2. A school cafeteria claims that students choose among 5 lunch options in equal proportions. In a random sample of $n=250$ lunches, the counts are shown below (expected count for each option is 50). A chi-square goodness-of-fit test gives p-value = 0.0008. What conclusion is appropriate at the $\alpha=0.05$ level? Observed/Expected counts (with p-value):

   Option	Observed	Expected
   1	70	50
   2	41	50
   3	56	50
   4	38	50
   5	45	50

   p-value = 0.0008
3. A school counselor fits a least-squares regression model to predict students' weekly study time (hours) from the number of extracurricular activities they participate in. The counselor claims that each additional activity is associated with a decrease in mean weekly study time. A 95% confidence interval for the slope is $(-1.8,\,-0.4)$ hours per activity. Is the claim supported by the confidence interval?
4. A marketing team models weekly sales ($y$) from weekly advertising spending in thousands of dollars ($x$) using 25 weeks. The team claims that spending has a positive linear effect on mean sales. A 95% confidence interval for the slope is $(-3.2, 14.5)$ (sales units per $$1{,}000). Is the claim supported by the confidence interval?
5. A teacher recorded homework completion (Completed or Not completed) and whether the student passed the unit test (Pass/Fail). To interpret the relationship using conditional distributions, identify the correct conditional comparison. Which comparison is appropriate?
6. A game developer says that 15% of players make an in-app purchase. Many random samples of $n=40$ players are taken and $\hat p$ is computed each time. Which statement about the sampling distribution is correct?
7. An engineer wants to test the effect of temperature (low vs. high) and catalyst type (C1 vs. C2 vs. C3) on the time to complete a chemical reaction. Each batch of chemicals can be used for only one run, and the engineer can randomly assign runs to conditions. The engineer wants to know whether the best catalyst depends on temperature. Which experimental design is most appropriate?
8. At a factory, the fill amount (in ounces) of individual bottles has population mean $\mu=20$ ounces and a fixed population standard deviation $\sigma$ (distribution may be skewed). Each hour, a quality engineer randomly selects $n=36$ bottles and records the sample mean fill amount $\bar{x}$. This process is repeated many times under identical conditions. Which statement about the sampling distribution of $\bar{x}$ is correct?
9. A researcher believes the mean amount of sleep for college students at a university is $\mu=7$ hours per night, but suspects it is actually less. A random sample of 100 students is surveyed, and a one-sample $t$ test is conducted with $H_0:\mu=7$ versus $H_a:\mu<7$ at $\alpha=0.05$. The p-value is 0.14. What conclusion is appropriate?
10. A restaurant chain claims that exactly 65% of customers order a drink with their meal. A market analyst believes the true proportion is greater than 65%. A random sample of 220 customers finds 156 who ordered a drink ($\hat{p}\approx 0.709$). Which hypotheses are appropriate for testing the analyst's belief?
11. A psychologist compares mean reaction time (in milliseconds) for participants who drank caffeinated coffee versus participants who drank decaffeinated coffee before a computer task. A sample of $n=22$ caffeinated participants had a mean reaction time of $\bar{x}=241.8$ ms, and a sample of $n=24$ decaffeinated participants had a mean reaction time of $\bar{x}=255.6$ ms. The psychologist claims caffeine leads to faster reactions (lower population mean reaction time). Which hypotheses are appropriate?
12. A botanist compares the mean height (in cm) of plants grown under LED light versus plants grown under fluorescent light. A random sample of $n=15$ LED-grown plants had a mean height of $\bar{x}=27.4$ cm, and a random sample of $n=16$ fluorescent-grown plants had a mean height of $\bar{x}=26.1$ cm. The botanist claims the population mean plant height differs between the two lighting types. Which hypotheses are appropriate?
13. A principal wants to estimate the mean number of hours of homework per week for all students at a high school. The principal takes a simple random sample of 80 students from the school roster using a random number generator. Which statement best describes the sample representativeness?
14. A delivery company audits 20 packages to see whether each arrives on time. A success is defined as an on-time delivery. Based on recent performance, the probability a package arrives on time is 0.81, and package outcomes are treated as independent. If $X$ is the number of on-time deliveries in the audit, which values correctly model this situation (identify $n$ and $p$)?
15. A runner records her mile times (in seconds) for 45 training runs. The dotplot shows a dense cluster from 430 to 470 seconds, with a few slower times at 520, 540, and 565 seconds, and no unusually fast times far below the cluster. Which statement best describes the distribution?
16. A fitness coach wants to estimate the mean resting heart rate (in beats per minute) of all clients at a gym. She takes a random sample of 36 clients and computes a 90% confidence interval for the population mean resting heart rate: $(68.2, 72.9)$ bpm. Which interpretation is correct?
17. A bank uses a predictive model built from a sample of past borrowers to infer whether a new applicant is likely to default. Based on the model's output, the bank must approve or deny the loan. Why is it important to consider potential error in this inference, given the impact on both the bank's losses and the applicant's access to credit?
18. A city health department asks: "Does sending text-message reminders increase flu vaccination rates?" The population is all adult residents of the city. The proposed plan is to recruit volunteers at a community health fair, then randomly assign those volunteers to receive weekly reminder texts or no texts, and compare vaccination within 2 months. Which aspect is most important to address before collecting data?
19. A card is drawn from a standard 52-card deck, the card is replaced, and the deck is shuffled. This process is repeated 6 times. Define a success as drawing a heart on a draw. Assume independence because of replacement and shuffling, and note that the probability of a heart on any draw is $\frac{13}{52}=0.25$. Let $X$ be the number of hearts drawn in the 6 draws. Which values correctly model this situation with a binomial distribution (identify $n$ and $p$)?
20. A city compared mean commute times (minutes) for residents who take public transit versus those who drive. From independent random samples, a 95% confidence interval for $(\mu_{\text{transit}}-\mu_{\text{drive}})$ was $( -6.5,\ 2.0)$. Which interpretation is correct?

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