Practice Test 22 - 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 science lab, measurement error (in millimeters) comes from two independent sources. Let $X$ be the error from the scale calibration with $\mu_X=0.2$ and $\sigma_X=0.6$. Let $Y$ be the error from the observer's reading with $\mu_Y=-0.1$ and $\sigma_Y=0.8$. Define $E=X+Y$, the total measurement error. Which statement about the combined variable is correct?
2. A university states that among all enrolled students, class year proportions are 27% first-year, 26% sophomore, 24% junior, and 23% senior. A random sample of 200 students is selected from the enrollment list and surveyed. A chi-square goodness-of-fit test is run at $\alpha=0.05$ with the results below. What conclusion is appropriate? Observed/Expected counts and p-value: - First-year: Observed 60, Expected 54 - Sophomore: Observed 45, Expected 52 - Junior: Observed 42, Expected 48 - Senior: Observed 53, Expected 46 - p-value = 0.072
3. A pharmaceutical company runs a clinical trial on a sample of patients and finds that a new drug appears to reduce symptoms more than a placebo. Regulators must decide whether to approve the drug for widespread use. Why is it important to consider potential error in the statistical inference from the trial when making this approval decision?
4. A plant scientist is testing two fertilizers (F1, F2) on tomato yield. The greenhouse has noticeable light differences between the north and south sides, and plants cannot be moved once placed. The scientist has 40 identical seedlings and can randomize which fertilizer each plant receives. Which experimental design is most appropriate?
5. A university wants to check whether the proportion of students who own a bicycle is 40%. A random sample of 250 students found that 112 own a bicycle. A one-proportion $z$ test was performed for $H_0: p=0.40$ versus $H_a: p \neq 0.40$ at $\alpha=0.01$. The $p$-value was 0.018, so the university failed to reject $H_0$. Which conclusion is appropriate?
6. A factory machine produces bolts, and an engineer inspects the next 50 bolts produced. Each bolt is classified as within tolerance (success) or out of tolerance (failure). The machine historically produces bolts within tolerance 92% of the time, and the engineer assumes independence and a constant probability for each bolt. Let $X$ be the number of bolts within tolerance. Which values correctly model this situation?
7. A genetic test is administered to 12 patients. A success is defined as the test returning a positive result for a particular marker, and for the population being tested the probability of a positive result is $0.10$ for each patient (assume independence). Which values of $n$ and $p$ correctly model the number of successes using a binomial distribution?
8. A university wants to estimate the mean number of hours per week that full-time students spend exercising. Two independent simple random samples of 40 full-time students are taken from the same semester's enrollment list. Sample A has a mean of 3.6 hours; Sample B has a mean of 4.4 hours. Why might the sample results differ?
9. A polling organization wants to estimate the population proportion $p$ of registered voters who support a bond measure. They randomly select 1,000 registered voters to call, but only those who answer and agree to respond are included. The point estimate is the sample proportion $\hat{p}$ among respondents. Which statement describes whether the estimate is biased?
10. A restaurant chain wants to compare customer satisfaction ratings across four locations. Independent random samples of customers were surveyed at each location and rated their experience as Satisfied or Not satisfied. A chi-square test of homogeneity was performed at $\alpha=0.05$. The observed counts and p-value are shown. What conclusion is appropriate?
11. A market analyst is comparing whether the distribution of preferred payment method (Credit, Debit, Cash, Mobile pay) is the same across two store locations (Mall, Downtown). The analyst takes separate random samples of customers from each location and records each customer's preferred payment method. The results are summarized in the two-way table. Which chi-square test is appropriate, and what are the correct hypotheses? Note: Multiple groups (locations) compared on one categorical response.
12. A beverage company completed an experiment to test whether a new packaging design increases purchase intent. The company recruited 300 online panelists (not randomly sampled from all consumers). Each participant was randomly assigned to view either the current package or the new package, then rated purchase intent on a 1–10 scale. The new-package group had a higher mean rating. Which conclusion is justified based on the design of this study?
13. A university expects the distribution of students' primary commute methods to be: 50% Car, 20% Bus, 15% Bike, 10% Walk, 5% Other. A random sample of 300 students reports: Car 141, Bus 72, Bike 39, Walk 33, Other 15. A chi-square goodness-of-fit test of $H_0$: the population distribution matches the expected proportions yields $p$-value = 0.27. Expected counts under $H_0$ are: Car 150, Bus 60, Bike 45, Walk 30, Other 15. What conclusion is appropriate at $\alpha=0.05$?
14. A delivery company models two independent random variables: $X$, the time (in minutes) to drive to a customer with $\mu_X=18$ and $\sigma_X=5$, and $Y$, the time (in minutes) to return to the warehouse with $\mu_Y=16$ and $\sigma_Y=4$. Define $R=X-Y$ (drive time minus return time). Which statement about $R$ is correct?
15. A restaurant manager claims that customers choose among three lunch specials (Pasta, Salad, Sandwich) in equal proportions. Over 200 randomly selected lunch orders on weekdays, the counts are recorded by special. The manager wants to test whether the observed distribution differs from the claimed equal distribution. Which inference procedure is most appropriate?
16. A researcher wants to study the relationship between weekly exercise time and resting heart rate among adults who belong to a particular gym. She records data from 60 gym members who volunteer to participate. The data show a negative association: members who report more weekly exercise tend to have lower resting heart rates. Which conclusion is supported by the data?
17. What is the standard deviation of C?
18. A school district claims that fewer than 30% of its high school students get at least 8 hours of sleep on school nights. A random sample of 200 students found that 48 reported getting at least 8 hours. A one-sample $z$ test for a population proportion was performed with hypotheses $H_0: p=0.30$ and $H_a: p<0.30$ at significance level $\alpha=0.05$. The test resulted in a $p$-value of 0.018, so the researchers rejected $H_0$. Which conclusion is appropriate?
19. Which of the following is the best conclusion based on this information?
20. A tech company wants to estimate the mean battery life (in hours) of a new smartphone model under a standard test. They randomly test 12 phones and compute a 98% confidence interval for the population mean battery life: $(9.6, 10.8)$ hours. Which interpretation is correct?

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