Practice Test 10 - Statistics and Probability

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

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1. A company reviewed employee records and found that employees with more years of experience tended to have higher annual salaries (a positive correlation). The company did not assign experience levels; it used existing records. Which statement is the most reasonable conclusion?
2. A school newsletter reports: "Students who use the library after school get higher grades." The newsletter surveyed 60 students who were already attending an after-school library program and found their average GPA was 3.4, compared with the schoolwide average GPA of 3.0 from last year's records. The survey did not include students who do not attend the library program. Which critique best evaluates the claim?
3. A survey asks adults about habits. Event $A$ is "a randomly selected respondent drinks coffee daily," and event $B$ is "a randomly selected respondent drinks tea daily." Results show $P(A)=0.62$, $P(B)=0.31$, and $P(A\cap B)=0.18$. What is $P(A\cup B)$, where "or" is inclusive (coffee or tea or both)?
4. A clinic reports $P(A\cap B)=0.06$ and $P(B)=0.20$, where event $A$ is "a patient has elevated blood pressure" and event $B$ is "a patient is over age 60." Based on the information given, which statement correctly interprets $P(A\mid B)$?
5. A teacher uses the model $\hat{y}=70+5x$ to predict a student's quiz score ($y$) from hours studied ($x$). For a student who studied $x=4$ hours, the residual was $-6$ (where $\text{residual}=y-\hat{y}$). For $x=4$, what does a residual of $-6$ mean in context?
6. The heights (in inches) of 10 plants in a classroom experiment were: 14, 15, 15, 16, 16, 17, 18, 18, 19, 22. Which statement correctly describes the distribution in a box plot of these data?
7. A theater manager wants to estimate the proportion of all tickets sold for next weekend's shows that will be purchased online. The manager randomly selects 150 tickets from the complete set of tickets sold so far for next weekend and finds that 90 were purchased online. What is a reasonable inference based on this sample?
8. A student council took a random sample of 120 students and recorded whether each student participates in a school club and whether they prefer morning or afternoon announcements. Using the two-way table, estimate the probability that a randomly selected student prefers morning announcements given that the student is in a club. Give your answer as a fraction. Two-way table (counts): - In a club & Morning: 42 - In a club & Afternoon: 18 - Not in a club & Morning: 30 - Not in a club & Afternoon: 30 Row totals: In a club 60, Not in a club 60 Column totals: Morning 72, Afternoon 48 Grand total: 120
9. A bookstore's records show $P(A\cap B)=0.15$ and $P(B)=0.50$, where event $A$ is "a customer buys fiction" and event $B$ is "a customer is a member of the rewards program." Based on the information given, what is $P(A\mid B)$? Give your answer as a percent.
10. A fair six-sided die is rolled once; outcomes are equally likely. Define the random variable $X$ as $X=\lvert\text{roll}-3\rvert$. The outcome-to-$X$ mapping is: $1\to 2$, $2\to 1$, $3\to 0$, $4\to 1$, $5\to 2$, $6\to 3$. Which table correctly represents the theoretical probability distribution of $X$?
11. A company tested a new phone setting designed to reduce screen time. In a randomized experiment, 200 users were randomly assigned to either have the setting turned on or left off for two weeks. At the end, each user's average daily screen time in minutes (quantitative) was recorded. The group with the setting turned on had lower average daily screen time. Can we conclude that turning on the setting causes a reduction in screen time? Why or why not?
12. A student recorded reaction times (in seconds) for 11 trials in a video game: 0.42, 0.44, 0.45, 0.46, 0.46, 0.47, 0.48, 0.49, 0.50, 0.52, 1.20 Which measure is most affected by the outlier 1.20 seconds?
13. A principal wants to estimate what percent of students prefer block scheduling over a traditional schedule. A link to an online poll is posted on the school's homepage, and students choose whether to respond. No incentives are provided. What is the best statement about what this design allows?
14. A phone company reports: "Battery Saver Mode doubles battery life." In a test, 10 employees used Battery Saver Mode and averaged 12 hours before needing a charge, while 10 other employees did not use it and averaged 6 hours. Employees chose which group to be in. Which critique best evaluates the claim?
15. A teacher randomly assigned 48 students to two note-taking methods while watching the same lecture: Treatment A (handwritten notes) and Treatment B (typed notes), with 24 students in each group. Afterward, students took a 20-question quiz. Results: Treatment A mean score = 16.1, Treatment B mean score = 15.6. Observed difference (A − B) = $0.5$ points. A randomization test was performed by shuffling the treatment labels 1,000 times under "no treatment effect" and recomputing (A − B). In the simulation, 410 of the 1,000 shuffled differences were at least as large as $0.5$. Which conclusion is most reasonable about the treatment effect?
16. A phone model has features that may overlap. Event $A$ is "a randomly selected phone has 5G," and event $B$ is "it has at least 256 GB of storage." The manufacturer reports $P(A)=0.60$, $P(B)=0.50$, and $P(A\cap B)=0.35$. What is the probability that $A$ or $B$ occurs, where "or" is inclusive (5G or 256 GB+ or both)?
17. A subscription app is deciding whether to show a retention offer to users who look likely to cancel. Their model produces a 'high-risk' flag. Based on past users: - $10\%$ of users would cancel within 30 days without any offer. - Of those who would cancel, the model flags $70\%$ as high-risk. - Of those who would not cancel, the model still flags $15\%$ as high-risk. The company can choose one policy: 1) Show the offer to all users. 2) Show the offer only to high-risk users. 3) Show the offer to no one. The company's goal is to target the offer so that most recipients are genuinely at risk of canceling (i.e., reduce wasted offers), while still reaching many would-be cancellers. Which strategy is most reasonable given the probabilities?
18. A company runs a two-step hiring process. First, an applicant passes the phone screen (event $A$) with probability $P(A)=0.50$. Then, given the applicant passed the phone screen, the probability they pass the onsite interview (event $B$) is $P(B\mid A)=0.40$. What is the probability that both events occur, $P(A\cap B)$?
19. A university wants to estimate the proportion of all undergraduate students who own a bicycle. It randomly selects 250 undergraduates from the registrar's list and surveys them. Which statement correctly describes what can be inferred from the sample?
20. A die is rolled once. Event $A$ is "the result is greater than 4," and event $B$ is "the result is even." Given that $P(A)=\tfrac{1}{3}$, $P(B)=0.50$, and $P(A\cap B)=0.20$, are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$).

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