Practice Test 9 - AP Statistics

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

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1. A farmer wants to estimate the mean weight of apples produced by a large orchard. On the same day, the farmer takes two independent random samples of 50 apples each from the orchard. Sample 1 has a mean weight of 182 grams; Sample 2 has a mean weight of 176 grams. Why might the sample results differ?
2. A school district randomly sampled 500 parents and found that 310 support switching to a later high school start time. A 98% confidence interval for the true proportion $p$ of all parents in the district who support the switch is $(0.57,\ 0.67)$. Which interpretation is correct?
3. A coin is flipped 30 times. The results, in order, are: H T H T H T H T H T H T H T H T H T H T H T H T H T H T H T. Under random flipping you would expect some clustering and occasional runs, not a perfectly alternating pattern across many flips. Is the pattern consistent with random behavior?
4. A streaming service claims that, among its users, time-of-day for starting a movie is distributed as: 20% Morning, 30% Afternoon, 35% Evening, 15% Night. A random sample of 200 movie starts yields: Morning 33, Afternoon 52, Evening 80, Night 35. A chi-square goodness-of-fit test of $H_0$: the population distribution matches the claim gives $p$-value = 0.006. Expected counts under $H_0$ are: Morning 40, Afternoon 60, Evening 70, Night 30. What conclusion is appropriate at $\alpha=0.05$?​
5. A school district wants to know whether the proportion of students who prefer taking tests on a computer differs between middle school and high school students. A random sample of 120 middle school students and 150 high school students is selected. Each student is asked whether they prefer computer-based tests (Yes/No). Which inference procedure is most appropriate to compare the two groups' preferences?
6. A college wants to test whether a new tutoring program improves final exam scores in Calculus I. From the 240 students enrolled, administrators assign students to tutoring or no tutoring based on whether they request extra help during the first week. At the end of the term, they compare mean final exam scores between the two groups. Which feature most limits the ability to draw a cause-and-effect conclusion?
7. What is the variance of X?
8. A quality-control technician inspects a shipment by selecting exactly 20 light bulbs at random from a large batch made on the same machine that day. Each bulb is classified as either defective (success) or not defective (failure). The technician records the number of defective bulbs found in the 20 inspections. Does this situation meet the conditions for using a binomial model?
9. A smartphone manufacturer claims the mean battery life under a standard test is $\mu=20$ hours. From a random sample of phones, a 99% confidence interval for $\mu$ is computed as $(19.2,\ 20.8)$ hours. Is the claim supported by the confidence interval?
10. A voter registry indicates that the true proportion of registered voters who are independent is $p=0.27$. A pollster repeatedly takes many random samples of size $n=120$ and computes $\hat p$ for each sample. Which statement about the sampling distribution of $\hat p$ is correct?
11. A teacher tracks the number of times, $X$, a student raises their hand during a randomly selected class period. The distribution of $X$ is discrete with mean $\mu_X=6.8$ and standard deviation $\sigma_X=2.5$. Which interpretation of the standard deviation is correct?
12. An online retailer summarized the delivery times (in days) for 100 randomly selected orders. The summaries were: mean $=5.6$ days, median $=5.0$ days, SD $=2.4$ days, $Q_1=4.0$ days, $Q_3=6.0$ days, minimum $=2$ days, maximum $=18$ days (five-number summary: $2, 4, 5, 6, 18$). Which interpretation is correct?
13. A school counselor summarizes the number of colleges to which 75 seniors applied. The five-number summary is: minimum $1$, $Q_1=3$, median $5$, $Q_3=7$, maximum $15$. A boxplot is created from these values. Which feature is consistent with the summary statistics?
14. A public health group took independent random samples from three neighborhoods (North, Central, South) and recorded whether residents are up-to-date on a flu vaccine (Yes/No). The results are shown in the two-way table. A chi-square test of homogeneity produced $p$-value $=0.0006$. What conclusion is appropriate at the 5% significance level?
15. A wildlife biologist models whether a tagged bird returns to a nesting site each year as independent with probability 0.70. To estimate the probability that in the next 5 years the bird returns in at least 4 of them, the biologist simulates 10,000 sequences of 5 years using random two-digit numbers (00–69 = return, 70–99 = not return). The simulation shows 5294 sequences with at least 4 returns. Which interpretation of the simulation results is correct?
16. A city transit agency claims the mean wait time for a bus on a certain route is $\mu=8$ minutes. Riders suspect the mean wait time is longer. A random sample of 50 wait times is recorded, and a one-sample $t$ test is conducted with $H_0:\mu=8$ versus $H_a:\mu>8$ at $\alpha=0.05$. The p-value is 0.051. What conclusion is appropriate?
17. A cafe compares the number of customers served per hour on weekdays versus weekends. The table shows summaries from several randomly selected hours. Which comparison is supported?

    Statistic	Weekday	Weekend
    Min	12	10
    $Q_1$	18	22
    Median	25	30
    $Q_3$	32	40
    Max	45	70

18. A bookstore tracks the number of books purchased, $X$, by a randomly selected customer. The distribution is discrete with mean $\mu_X=1.8$ books and standard deviation $\sigma_X=0.9$ books. Which interpretation of the mean is correct?
19. A student randomly selects one card from a standard 52-card deck and records the outcome. Let event $A$ be "the card is a heart" and event $B$ be "the card is a king." A king of hearts exists in the deck, so the two events can occur on the same draw. Are the two events mutually exclusive?
20. A researcher measures the reaction time (in milliseconds) of individual adults on a computer task. The population mean is $\mu=250$ ms and the population standard deviation is $\sigma=60$ ms; the population distribution is unimodal but not perfectly symmetric. The researcher repeatedly takes simple random samples of size $n=10$ and computes the sample mean reaction time $\bar{x}$. Which statement about the sampling distribution of $\bar{x}$ is correct?

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