Practice Test 26 - 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 teacher wants to know whether using a spaced-practice app improves quiz scores. She randomly assigns students in her class to either (1) use the app for 10 minutes per day for two weeks or (2) study as they usually do for two weeks. Then everyone takes the same quiz. Which statement correctly distinguishes random sampling and random assignment in this study?
2. A survey records how many cups of coffee a randomly selected student drank yesterday. Let $X$ be the number of cups. The distribution is: $P(X=0)=0.2$, $P(X=1)=0.5$, $P(X=2)=0.2$, $P(X=3)=0.1$. What is the expected value of $X$ (in cups)? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)
3. The weights of packages shipped by a warehouse are roughly normal for a certain product line. Package weights have mean $\mu=20$ lb and standard deviation $\sigma=3$ lb. Approximately what percent of packages weigh below 17 lb? (Use the empirical rule.)
4. A library recorded 90 teen visits by day type and whether the student checked out a book. Based on the two-way table, what is the conditional relative frequency (as a decimal) of Checked out a book = Yes given that the visit was on a Weekend?
5. A school randomly assigned 60 students to use either Treatment A (a spaced-practice study plan) or Treatment B (a cramming study plan) for one week before a quiz. Treatment A had $n=30$ with a mean quiz score of 82 points; Treatment B had $n=30$ with a mean quiz score of 74 points. The observed difference in means was $82-74=8$ points (A − B). To test whether this difference could be due to chance under "no treatment effect," the teacher performed a randomization test by shuffling the treatment labels 1,000 times and recalculating (A − B) each time. In the simulation, 9 of the 1,000 shuffled differences were at least as large as $8$ points. Which conclusion is most reasonable about the treatment effect?
6. Using the outcomes listed for a spinner with 6 equal sections labeled $S=\{R1,R2,B1,B2,G1,G2\}$ (R=red, B=blue, G=green), define event $A$ as "the outcome is red" and event $B$ as "the outcome has number 2." Which expression represents $A\cup B$ (A or B)?
7. Based on the sample space shown for choosing one letter, $S=\{A,B,C,D,E,F\}$, define event $A=\{x\in S\mid x\text{ is a vowel}\}$. Which set of outcomes represents event $A$?
8. In a student club census, event $A$ is "a student is in the debate club" and event $B$ is "a student is in the robotics club." The probabilities are $P(A)=0.22$, $P(B)=0.31$, and $P(A\cap B)=0.09$. What is the probability that $A$ or $B$ occurs (inclusive or)?
9. Four players—Lia, Mo, Nia, and Pax—are choosing who gets to be captain. They consider several methods. Fair means each player has probability $\tfrac14$ of being selected, even if the "most deserving" person doesn't always win. Which procedure is fair?
10. A shopper is choosing between two membership plans for a store. Compare the expected net savings per month (savings minus membership fee) over many months. Strategy A (Basic): - Monthly fee: $5 - With probability 0.40, you save $20 that month - With probability 0.60, you save $0 that month Strategy B (Plus): - Monthly fee: $12 - With probability 0.25, you save $50 that month - With probability 0.75, you save $0 that month Which option would result in a higher average net savings per month over many months?
11. A library wants to estimate the population parameter $p$, the proportion of all library visitors who prefer e-books over printed books. A random sample of $n=120$ visitors was surveyed, and the sample statistic was $\hat{p}=0.35$. Students ran a simulation of repeated random sampling (5,000 samples of size 120) using a model where the true proportion is 0.35. They found that about 95% of the simulated sample proportions were within 0.08 of 0.35. What does the margin of error mean in context?
12. Using the outcomes listed in $S=\{0,1,2,3,4,5,6,7,8,9\}$ for a random experiment where one digit is selected, let $A=\{x\in S\mid x\text{ is a multiple of }3\}$ and let $B=\{x\in S\mid x\text{ is even}\}$. Which set of outcomes represents $A\cup B$ (A or B)?
13. A company inspects products for features. Event $A$ is "a randomly selected product is waterproof," and event $B$ is "it has wireless charging." The data show $P(A)=0.55$, $P(B)=0.40$, and $P(A\cap B)=0.25$. What is the probability that $A$ or $B$ occurs, where "or" is inclusive (waterproof or wireless charging or both)?
14. There are 7 runners in a race. Gold, silver, and bronze medals are awarded. Because medals are different, order matters. What is the probability that two particular runners, A and B, both win medals (in any order)?
15. A retailer suspects some online orders are fraudulent. Historically, 2% of orders are fraudulent. Two automated rules are being considered: - Rule High-Sensitivity: flags 95% of fraudulent orders, but flags 10% of legitimate orders. - Rule High-Precision: flags 70% of fraudulent orders, but flags 1% of legitimate orders. Flagged orders are held for manual review, while unflagged orders ship immediately. The retailer's goal is to minimize the probability that a shipped (unreviewed) order is fraudulent, even if that increases the number of reviews. Which strategy is most reasonable given the probabilities?
16. A librarian wants to test whether a reminder text message increases the rate of on-time book returns. Each student who checks out a book is randomly assigned to either receive a reminder text 2 days before the due date or receive no text. The librarian then records whether each book is returned on time. Which type of study is described?
17. A delivery company is choosing between two routing software packages. For each day, exactly one traffic condition occurs, producing the listed net profit (after all costs) for that day. Strategy A: - Net profit $900 with probability 0.50 - Net profit $300 with probability 0.30 - Net loss $200 with probability 0.20 Strategy B: - Net profit $700 with probability 0.70 - Net profit $100 with probability 0.20 - Net loss $500 with probability 0.10 Which strategy has the greater expected value (higher average net profit per day) over many days?
18. A water tank is being filled and a model for the amount of water is $y = 6x - 10,$ where $x$ is time since filling began (minutes) and $y$ is the amount of water in the tank (liters). Is the $y$-intercept meaningful in context? Choose the best statement.
19. A wildlife volunteer recorded the number of minutes it took to spot a particular bird on 13 different mornings: 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 12, 25 Which measure of spread is most increased by the 25-minute morning?
20. At a school, two events are defined: Event A = "a student earns an A in English," and Event B = "a student is in the yearbook club." A student says, "If you're in yearbook club, you'll get an A in English." The data only show that A's are more common among yearbook club members than among nonmembers. Which statement best describes whether A and B are independent?

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