Practice Test 51 - AP Statistics

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

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1. A public health department wants to estimate the proportion of adults in a county who received a flu shot this season. Two independent simple random samples of 200 adults are drawn from the same county registry. Sample 1 reports 62% vaccinated; Sample 2 reports 58% vaccinated. Why might the sample results differ?
2. A city health department wants to know whether a new text-message reminder system increases the percentage of patients who arrive on time for appointments. Over one month, the department randomly selected 400 patients from all patients with appointments and randomly assigned 200 to receive text reminders and 200 to receive no reminders. The on-time rate was higher in the reminder group. Which conclusion is justified based on the design of this study?
3. A health clinic believes that more than 25% of adults in its county have not received a flu shot this season. A random sample of 400 adults found 118 had not received a flu shot. A one-sample $z$ test for a population proportion was performed with $H_0: p=0.25$ versus $H_a: p>0.25$ at $\alpha=0.05$. The test produced a $p$-value of 0.009, so $H_0$ was rejected. Which conclusion is appropriate?
4. What is the probability that a randomly selected player scored more than 1 goal but no more than 4 goals?
5. A company tracks the number of minutes customers spend on hold. For individual calls, the mean is $\mu=6$ minutes and the standard deviation is $\sigma=3$ minutes, and the distribution is right-skewed. A manager repeatedly selects many random samples of $n=40$ calls and computes the sample mean hold time $\bar{x}$. Which statement about the sampling distribution of $\bar{x}$ is correct?
6. A student claims a coin is fair and flips it 40 times, recording the sequence. The results include a long streak of 12 heads in a row at one point, with the remaining flips a mix of heads and tails. Is the pattern consistent with random behavior? Under randomness, you would expect occasional long streaks to appear by chance, especially in longer sequences, even if outcomes are independent.
7. A wildlife biologist estimates that 35% of a large population of turtles have a tracking tag. She repeatedly takes many random samples of $n=25$ turtles and computes $\hat p$. Which statement about the sampling distribution is correct?
8. A company compares delivery times (minutes) for two shipping options: Standard and Express. The side-by-side summaries below were computed from similar sample sizes. Which comparison is supported?

   Statistic	Standard	Express
   Min	28	18
   $Q_1$	40	30
   Median	55	42
   $Q_3$	75	55
   Max	160	120
   Outliers noted	several high values above 140	none noted

9. A fitness app recorded the number of minutes exercised per week for a random sample of 25 users. The five-number summary was: min $=30$, $Q_1=75$, median $=110$, $Q_3=150$, max $=240$. A boxplot is drawn from these values. Which feature is consistent with the summary statistics?
10. A factory reports that 8% of items are defective. Of all items, 30% are produced by Machine 1, and 3% of all items are both defective and produced by Machine 1. Let Event A be "item is defective" and Event B be "item is produced by Machine 1." Which statement about Events A and B is correct?
11. A sports scientist studies whether weekly training hours ($x$) can predict 5K race time in minutes ($y$) for competitive runners. A least-squares regression of $y$ on $x$ is planned, and the scientist claims that increasing training hours is associated with faster (smaller) race times in the population. Which hypotheses are appropriate for testing the scientist's claim about the slope of the population regression line?
12. A class uses a random number generator to select one student each day from a roster of 25. Over 25 days, the teacher records the selected student number (1–25). The list shows that student 12 is selected on 6 different days, while student 7 is never selected. Under random behavior, you would expect some students to be selected multiple times and some possibly not at all. Is the pattern consistent with random behavior?
13. A manufacturer produces resistors with mean resistance $\mu=100\ \Omega$ and population standard deviation $\sigma=8\ \Omega$. Individual resistances are approximately normal. A quality engineer repeatedly selects simple random samples of size $n=16$ and records the sample mean resistance $\bar{x}$. Which statement about the sampling distribution of $\bar{x}$ is correct?
14. A tech company sampled 50 employees and recorded hours of weekly training ($x$) and number of coding errors found in review ($y$). A least-squares regression of $y$ on $x$ was fit. The slope was tested using $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$, and the p-value was 0.0008 with a negative estimated slope. Using $\alpha=0.05$, the company wants to draw a conclusion. What conclusion is appropriate?
15. Given the information provided, what was the overall total number of students surveyed who participate in sports?
16. A wildlife biologist wants to estimate the mean length of a certain fish species in a large lake during the summer. Two independent simple random samples of 30 fish are caught and measured from the same lake during the same week. Sample 1 has a mean length of 18.4 cm; Sample 2 has a mean length of 20.1 cm. Why might the sample results differ?
17. A school surveyed 200 students about whether they participate in a sport and whether they prefer morning or afternoon classes. Under the assumption that sport participation and class-time preference are independent, which expression calculates the expected count for the Sport & Morning cell?
18. A veterinarian studies whether dog breed group is associated with adult body mass. For each of 60 adult dogs at a clinic, the vet records breed group (coded 1 = toy, 2 = herding, 3 = sporting, 4 = working) and the dog's mass in kilograms. The observational units are the dogs. Which classification is correct?
19. A teacher compares mean quiz scores for students who studied with flashcards versus students who studied by rereading notes. Two independent random samples were taken (flashcards: $n=16$; rereading: $n=18$). A two-sample $t$ test for $\mu_{flash}-\mu_{read}$ used $H_0:\mu_{flash}-\mu_{read}=0$ and $H_a:\mu_{flash}-\mu_{read}>0$. The p-value was 0.20. At the 0.05 significance level, what conclusion is appropriate?
20. A teacher flips a coin 40 times and records the results. The sequence includes many short runs and alternations, but also has one stretch of 8 heads in a row. Is the pattern consistent with random behavior?

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