Practice Test 28 - Statistics and Probability

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

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1. Using the outcomes listed, $S=\{\text{cat},\text{dog},\text{fish},\text{bird}\}$ for randomly selecting one sticker, let event $A$ be "selects a pet that lives in water" and let event $B$ be "selects a pet that can fly." Which set represents $A\cup B$ (A or B)?
2. A lab measured the pH of 13 water samples from the same source: 7.1, 7.1, 7.2, 7.2, 7.2, 7.3, 7.3, 7.3, 7.4, 7.4, 7.4, 7.5, 8.6 Which conclusion about the data is best supported?
3. The lifetimes of a certain type of light bulb (in hours) are roughly normal based on historical quality-control data (symmetric and bell-shaped). Lifetimes are modeled as $N(\mu=1200,\,\sigma=100)$. Approximately what percent of bulbs last between $1100$ and $1300$ hours? (Use the empirical rule.)
4. A student plays a carnival game many times and wants the higher average profit (expected value) per play. Strategy A: Pay $4 to play. Outcomes: with probability 0.20 you win $20; with probability 0.80 you win $0. Strategy B: Pay $4 to play. Outcomes: with probability 0.60 you win $8; with probability 0.40 you win $0. Which strategy has the greater expected value (average profit) over many repetitions?
5. School context: Two events are defined. Event A: a student turned in the history project on time. Event B: a student came to the optional review session. The teacher notices that the on-time rate is about the same for students who came to the review session as it is for the class overall. Which statement best describes whether A and B are independent?
6. A district wants to compare two tutoring formats (in-person vs online) for improving algebra test scores. They randomly select 10 algebra classes from the district and then randomly assign the selected classes (not individual students) to use in-person tutoring or online tutoring for 6 weeks. Which statement correctly distinguishes random sampling and random assignment in this study?
7. A coffee shop chain wants to estimate the mean amount (in dollars) spent per transaction across all transactions at one location on Saturday. The manager randomly selects 30 transactions from that Saturday's receipts and computes the sample mean spending. What population parameter is being estimated?
8. A library recorded whether 140 students checked out a fiction book or a nonfiction book, and whether they visited the library during lunch or after school. The two-way table shows the counts. Which statement best describes the association between time of visit and type of book checked out?

   	Fiction	Nonfiction	Total
   Lunch	30	20	50
   After school	70	20	90
   Total	100	40	140

9. A coach modeled the relationship between practice time $x$ (hours) and free-throw percentage $y$ using $\hat{y}=60+2x$. Residuals were calculated as $\text{residual}=y-\hat{y}$. Which statement best describes how well the model fits the data based on the residuals shown?
10. A teacher tested two ways to practice vocabulary. She randomly assigned 48 students to Treatment A (flashcards) and 48 students to Treatment B (practice sentences). At the end of the week, each student took a 30-word quiz. Treatment A had a mean of 24.1 correct and Treatment B had a mean of 24.4 correct, so the observed difference in means was $24.1-24.4=-0.3$ (A − B). A randomization test was run by shuffling the A/B labels 1000 times under "no treatment effect." In 1000 shuffles, 462 produced a difference in means of at most $-0.3$ (A − B) (i.e., A was at least 0.3 lower than B). Based on the randomization test, is the observed difference surprising under no effect?
11. At a university, event $A$ is "a randomly selected student is majoring in biology" and event $B$ is "a randomly selected student is taking calculus this term." The registrar reports $P(A)=0.30$, $P(B)=0.42$, and $P(A\cap B)=0.12$. What is the probability that a student is majoring in biology or taking calculus (inclusive or)?
12. School context: Two events are defined. Event A: a student completed the reading assignment. Event B: a student was absent at least one day this week. The teacher reports: "Among students who were absent at least one day, 40% completed the reading assignment. Among students who were not absent at all, 70% completed the reading assignment." Which statement best describes whether A and B are independent?
13. A community center wants to estimate the population parameter $p$, the proportion of all members who attend at least one class per week. A random sample of $n=100$ members is surveyed and the sample statistic is $\hat{p}=0.37$. Students use simulation/repeated sampling: they take 1000 simulated random samples of size 100 and compute $\hat{p}$ each time. They find that about 95% of the simulated sample proportions fall between 0.29 and 0.45. What does the margin of error mean in this context?​
14. A city models the relationship between outdoor temperature $x$ (\u00b0C) and daily electricity use $y$ (MWh) with $\hat{y}=120-2x$. A residual plot shows a funnel shape: residuals are close to 0 for low $x$ but become much more spread out (both positive and negative) as $x$ increases, while still centered around the 0 line. What does the residual pattern suggest about the model choice?
15. A small bookstore recorded the number of customers per hour on a weekday: 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 14, 29 Which measure is most affected by the outlier value of 29 customers?
16. The quiz scores (out of 20) for 13 students were: 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17. Which plot best represents the data as a dot plot on a number line?
17. A store wants to estimate the population parameter $p$, the proportion of all customers who use self-checkout. A random sample of 500 customers found 310 used self-checkout, so the sample statistic is $\hat p = 310/500 = 0.62$ (62%). Students ran a simulation of 1,000 repeated random samples of size 500 and found that about 95% of simulated sample proportions were between 0.58 and 0.66. What margin of error is supported by the simulation for estimating the population parameter $p$? (Express your answer as a percent.)
18. A basketball player's made free throws are recorded over 30 practice shots. The observed outcomes were: miss (0 points) occurred 8 times and make (1 point) occurred 22 times. Let $X$ be the points scored on a randomly selected shot (0 or 1). Using relative frequencies from the observed data as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
19. A school surveyed 90 students about whether they prefer science or history, and whether they are in an honors class or a regular class. The two-way table shows the counts. Which value represents the joint relative frequency (as a decimal) of a student who is in an honors class and prefers science?

    	Prefer science	Prefer history	Total
    Honors	24	6	30
    Regular	30	30	60
    Total	54	36	90

20. A cafeteria tracked the waiting times (in minutes) for 16 students: 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10, 11. Which statement correctly describes the distribution when shown in a histogram with 2-minute bins: 2–3, 4–5, 6–7, 8–9, 10–11?

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