Practice Test 7 - AP Statistics

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

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1. A researcher collected data from 35 commuters on commute distance ($x$, miles) and commute time ($y$, minutes). A least-squares regression of $y$ on $x$ was fit. The slope test $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ yielded a p-value of 0.0002 and a positive estimated slope. At $\alpha=0.05$, the researcher rejected $H_0$. What conclusion is appropriate?
2. A political analyst claims that the long-run distribution of party affiliation in a county is 48% Independent, 32% Democrat, and 20% Republican. A random sample of 250 registered voters is selected and asked their affiliation. Observed counts table: Independent 110, Democrat 88, Republican 52 Claimed distribution: (0.48, 0.32, 0.20). Which hypotheses are appropriate for a chi-square goodness-of-fit test?
3. A quality inspector counts the number of defective items in a randomly selected box of 5 items. Let $X$ be a discrete random variable defined as the number of defective items in the box. The probability distribution of $X$ is shown in a table. Which statement correctly interprets the random variable?
4. A manufacturer tests whether the mean fill weight of cereal boxes is less than the labeled 500 g. They test $H_0: \mu=500$ vs. $H_a: \mu<500$ at $\alpha=0.05$. The p-value is 0.22, so they fail to reject $H_0$ and conclude there is not convincing evidence the mean fill weight is less than 500 g. In reality, the true mean fill weight is $\mu=492$ g. Which type of error was made?
5. A technology teacher wants to know whether device type used for homework is associated with whether assignments are submitted on time. A random sample of students is taken and categorized by primary device (Phone, Tablet, Laptop) and submission status (On time, Late). The two-way table below summarizes the data. A chi-square test of independence gives $p=0.006$. What conclusion is appropriate at the $\alpha=0.05$ level? Two-way table (counts):

   Device	On time	Late
   Phone	35	25
   Tablet	38	12
   Laptop	70	20

6. A streaming service tests whether more than 60% of users prefer a new interface. They test $H_0: p=0.60$ versus $H_a: p>0.60$ at $\alpha=0.05$. The test result is to reject $H_0$, so the company rolls out the interface, claiming a majority prefers it. In truth, exactly 60% of users prefer the new interface. Which type of error was made?
7. A restaurant owner claims that customers pay using 4 methods in these proportions: 0.50 credit card, 0.20 debit card, 0.20 cash, 0.10 mobile pay. A random sample of 150 transactions is recorded and a chi-square goodness-of-fit test is conducted for $H_0$: the population payment-method distribution matches the claim versus $H_a$: it does not. The p-value is 0.95. The observed and expected counts are shown. What conclusion is appropriate at the $\alpha=0.05$ level?
8. A gym manager randomly sampled 20 treadmills of the same model and recorded the maximum speed (in mph) each could reach. A 90% confidence interval for the population mean maximum speed was $(11.6, 12.4)$ mph. Which interpretation is correct?
9. A community center randomly sampled adults and recorded whether they have a gym membership (Yes/No) and their transportation type to work (Car, Public transit, Walk/Bike). The two-way table is shown. A chi-square test of independence gave $p$-value $=0.12$. What conclusion is appropriate at the 5% significance level?
10. Let $X$ be the discrete random variable representing the number of customers who enter a small coffee shop between 8:00 and 8:10 A.M. Over many days, the distribution of $X$ has mean $\mu_X=6.2$ customers and standard deviation $\sigma_X=1.5$ customers. Which interpretation of the mean is correct?
11. A teacher records scores (out of 10) on a short quiz for 50 students, graded consistently using an answer key. The dotplot shows the distribution. Dotplot of quiz score 2 | • 3 | •• 4 | ••• 5 | •••• 6 | ••••• 7 | ••••••• 8 | •••••••••• 9 | •••••••••••• 10| ••••••• Which feature is most evident?
12. A community clinic asked a random sample of 100 patients which appointment type they scheduled (categorical variable). The bar chart shows counts: Checkup 33, Vaccination 21, Sick visit 27, Follow-up 14, Other 5. Which statement is supported by the graph?
13. A farm cooperative claims the mean weight of its apples is 180 grams. A grocery chain suspects the cooperative's apples are heavier on average. A random sample of $n=48$ apples has mean weight $ overline{x}=186$ grams. Which hypotheses are appropriate for testing the grocery chain's claim about the population mean apple weight, $\mu$?
14. A bag contains 5 white and 5 black chips. Two chips are drawn with replacement. Let Event A be "the first chip is white" and Event B be "the second chip is white." Which statement about Events A and B is correct?
15. A manager recorded the number of minutes late for each of 30 employees on a particular Monday. The dotplot below shows the distribution of minutes late (0 means on time). Which statement best describes the distribution?
16. A school compared the proportion of students who prefer online homework in two grades. In a random sample, 84 of 150 ninth-graders and 72 of 160 tenth-graders said they prefer online homework. A 95% confidence interval for the difference in population proportions, $p_9 - p_{10}$, is $(0.02,\ 0.20)$. Which interpretation is correct?
17. A coach studies whether the number of free-throw practice shots per week ($x$) predicts free-throw percentage in games ($y$). The coach claims that more practice shots increases free-throw percentage in the population. Which hypotheses are appropriate for testing the coach's claim about the population regression slope?
18. A nutritionist compares mean sodium intake (mg/day) for adults who follow Diet A versus Diet B. Independent random samples were taken from each diet group. A two-sample $t$ test was conducted for $H_0: \mu_A-\mu_B=0$ versus $H_a: \mu_A-\mu_B\ne 0$. The p-value was 0.41. At the 0.05 significance level, what conclusion is appropriate?
19. A city health department samples 60 households and estimates the mean weekly amount of trash produced per household. A 95% confidence interval for the population mean is $(24.1, 27.6)$ pounds. The director claims that the true mean weekly trash per household is 25 pounds. Is the claim supported by the confidence interval?
20. A movie theater surveys a randomly selected customer about how many snacks they purchased. Define a discrete random variable $S$ as the number of snacks purchased (0, 1, 2, or 3). The probability distribution of $S$ is shown in the table. Which statement correctly interprets the random variable?

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