Practice Test 61 - AP Statistics

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

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1. A political analyst claims that 52% of adults in a state approve of the governor. A random sample of 500 adults finds 245 who approve. A one-sample $z$ test for a population proportion is performed with $H_0: p=0.52$ and $H_a: p\ne 0.52$ at $\alpha=0.05$. The $p$-value is 0.09, so the null hypothesis is not rejected. Which conclusion is appropriate?
2. A biology teacher claims that pea plants in a lab activity will show a 3:1 ratio of purple flowers to white flowers (75% purple, 25% white). A student grows 120 plants and records the observed counts below. Which hypotheses are appropriate for a chi-square goodness-of-fit test of the teacher's claim?
3. A student guesses answers on multiple-choice questions until getting the first correct answer. Assume each guess is independent and the probability of a correct guess is constant from trial to trial. The student records the number of guesses until the first correct answer. Why is a geometric model appropriate?
4. A cereal company claims the mean amount of cereal in its boxes is $\mu=18$ oz. Inspectors take a random sample of boxes and compute a 95% confidence interval for $\mu$ as $(17.95,\ 18.10)$ oz. Is the claim supported by the confidence interval?
5. A fitness center manager is researching which class type members most want added to the schedule. From the population of current members, a random sample of 160 members is surveyed and each member selects a preferred new class type (a categorical variable). The distribution is shown below. Which statement is supported by the data?

   Preferred new class type	Frequency	Relative frequency
   Yoga	52	0.325
   Strength training	44	0.275
   Cycling	36	0.225
   Dance	28	0.175

6. A wildlife biologist studies the number of insect bites a hiker gets on a 2-hour trail. The population distribution of bites is strongly right-skewed because most hikers get few bites but a few get many. She takes many random samples of $n=60$ hikers and records the sample mean number of bites, $\bar{x}$, for each sample. She considers the sampling distribution of $\bar{x}$. Why is the sampling distribution approximately normal?
7. A restaurant chain wants to estimate the proportion of all customers who would recommend the restaurant. Receipts include a QR code inviting customers to complete a short survey for a chance to win a $50 gift card. The chain summarizes the survey responses and reports them as representative of all customers. Which issue most threatens the validity of this approach?
8. What proportion of patients in the sample have Type O blood?
9. To convert a relative frequency table back to a frequency table, what additional piece of information is required?
10. A website reports that 10% of visitors click on a particular ad. A researcher repeatedly takes random samples of $n=200$ visitors and calculates $\hat p$, the sample proportion who clicked. Which statement about the sampling distribution of $\hat p$ is correct?
11. A researcher compares mean daily screen time (hours) for teenagers in Urban areas versus Rural areas. Independent random samples were taken (Urban: $n=45$; Rural: $n=43$). A two-sample $t$ test for $\mu_U-\mu_R$ used hypotheses $H_0:\mu_U=\mu_R$ and $H_a:\mu_U\ne\mu_R$. The p-value was 0.051. Using $\alpha=0.05$, what conclusion is appropriate?
12. A principal compares the number of absences per student in a semester for 9th graders and 12th graders. The side-by-side summaries below come from random samples. Which comparison is supported?

    Statistic	9th grade	12th grade
    Min	0	0
    $Q_1$	1	1
    Median	3	2
    $Q_3$	6	4
    Max	25	12
    Outliers noted	several high values above 15	none noted

13. A hospital tested whether a new discharge checklist reduces 30-day readmissions. Over three months, nurses on each shift alternated: on even-numbered days they used the new checklist for all discharged patients, and on odd-numbered days they used the old process for all discharged patients. The hospital compared 30-day readmission rates for patients discharged on even vs. odd days. Which conclusion is justified based on the design of this study?
14. A botanist compares mean plant height (cm) after 6 weeks for plants grown under LED light versus fluorescent light. A random sample of $n=22$ LED-grown plants had mean $\bar{x}=34.5$ cm, and a random sample of $n=21$ fluorescent-grown plants had mean $\bar{x}=33.8$ cm. The claim is that LED lighting leads to a higher population mean height than fluorescent lighting. Which hypotheses are appropriate?
15. A company wants to answer: "Does a new 4-day workweek policy increase employee productivity?" The population is all employees at the company. The plan is to let each department vote on whether to adopt the 4-day schedule next quarter, then compare average productivity between departments that adopt and those that do not. Which aspect is most important to address before collecting data?
16. A call center dials phone numbers one at a time until reaching the first person who answers. Assume each dial is independent and the probability a dial is answered is constant. The supervisor records the number of dials until the first answered call. Why is a geometric model appropriate?
17. A farmer wants to compare two fertilizers (F1 and F2) on crop yield. The field has a noticeable slope, and yield tends to be higher at the bottom due to moisture. The farmer can divide the field into strips running across the slope so that each strip is relatively uniform in moisture. Each strip can be split into two equal plots, one for each fertilizer. Which experimental design is most appropriate to control for the slope effect?
18. A botanist measures, for 24 plants of the same species, the amount of fertilizer applied each week (x, in grams) and the plant's height after 6 weeks (y, in centimeters). The botanist wants to know whether fertilizer amount and height appear to be related. Do the data suggest the variables are related?
19. A car dealership summarized the selling prices (in thousands of dollars) of 30 used cars sold last month. The five-number summary was: min $=8.5$, $Q_1=10.0$, median $=12.5$, $Q_3=15.0$, max $=22.0$. A boxplot is drawn. Which feature is consistent with the summary statistics?
20. Let $X$ be the discrete random variable representing the number of text messages a student receives during a 50-minute class period. For this student, $\mu_X=12.0$ and $\sigma_X=4.0$. Which interpretation of the standard deviation is correct?

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