Practice Test 42 - 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 counselor wants to estimate the proportion of seniors who have completed a college application. The counselor surveys the first 80 seniors who arrive at a morning assembly and asks whether they have submitted at least one application. What is the main issue with using this sample?
2. A player repeatedly chooses between two prize wheels. The player wants the higher expected winnings per spin (in dollars). Strategy A: With probability 0.70, win $3; with probability 0.20, win $8; with probability 0.10, win $0. Strategy B: With probability 0.50, win $4; with probability 0.25, win $10; with probability 0.25, win $1. Which strategy has the greater expected value over many repetitions?
3. Six students—A, B, C, D, E, and F—need a fair way to choose who cleans the board today. Fair means each student has probability $1/6$ of being chosen. Which method gives each student an equal chance?
4. A game uses a fair spinner with 6 equally likely outcomes labeled $1,2,3,4,5,6$. Define the random variable $X$ as the payoff (in dollars) where outcomes $1$ or $2$ pay $0$, outcomes $3$ or $4$ pay $2$, outcome $5$ pays $5$, and outcome $6$ pays $8$. What is the expected value of $X$ based on this theoretical model?
5. A raffle ticket has the following net winnings (in dollars). Let $X$ be the net winnings from buying one ticket. $P(X=-2)=\tfrac{3}{4}$ and $P(X=6)=\tfrac{1}{4}$. What is the expected value of $X$ (in dollars)? (Note: the expected value may not be a possible outcome.)
6. A city library wants to estimate the proportion of all registered library card holders in the city who prefer e-books over printed books. The library uses a computer to randomly select 120 card holders from its registration list and surveys them; 48 say they prefer e-books. What population parameter is being estimated?
7. Six different photos are to be displayed in a row on a shelf. Order matters because different left-to-right orders are different displays. There are $6!$ equally likely arrangements. What is the probability that a particular photo (Photo X) is placed at one of the two ends of the row?
8. Two cards are drawn without replacement from a standard deck of 52 cards. Define the random variable $X$ as the number of aces drawn. Which table correctly represents the probability distribution of $X$?
9. School context: Two events are defined. Event A: a student joins the Yearbook Committee. Event B: a student takes an art class this semester. The advisor says, "Students in art class join yearbook more often than students not in art class." Which statement best describes whether A and B are independent?​​
10. A gamer can choose between two loot-box strategies each time they play, and they care about average profit over many plays (profit can be negative). Strategy A (costs $4 to open): - 0.60 probability of getting an item worth $2 - 0.30 probability of getting an item worth $6 - 0.10 probability of getting an item worth $20 Strategy B (costs $4 to open): - 0.80 probability of getting an item worth $3 - 0.15 probability of getting an item worth $8 - 0.05 probability of getting an item worth $25 Which strategy has the greater expected value (average profit) over many repetitions?
11. A student council wants to know what percentage of students support adding a longer lunch period. Council members stand near the cafeteria entrance at lunch and ask the first 80 students who walk in whether they support the change. No policy is implemented as part of the study. Which statement best describes the study and the role of randomization?
12. A fair six-sided die is rolled once. The sample space is $\{1,2,3,4,5,6\}$ with all outcomes equally likely. Define the random variable $X$ as the number of points you win where you win 2 points if the roll is even and 0 points if the roll is odd. (So outcomes $\{2,4,6\}$ map to $X=2$ and outcomes $\{1,3,5\}$ map to $X=0$.) What is the expected value of $X$ based on this theoretical model?
13. A prize wheel has 5 equal sections labeled $0, $0, $1, $2, and $5. You spin once. Let $X$ be the amount of money won. Which table correctly represents the probability distribution of $X$?
14. Two-step experiment with replacement: A bag contains many marbles of different colors. You draw one marble, record whether it is red, replace it, then draw again and record whether it is striped. Event $A$ is "the first draw is red," and event $B$ is "the second draw is striped." Given that $P(A)=0.50$, $P(B)=\tfrac{1}{5}$, and $P(A\cap B)=0.08$, are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$).
15. A student measured the mass (in grams) of 10 apples: 148, 150, 151, 152, 152, 153, 154, 156, 158, 160. Which statement correctly describes the distribution shown by a box plot of these data?
16. A student collected paired data on the number of pages read (pages) and the time spent reading (minutes) for several reading sessions. Technology reported a correlation coefficient of $r = 0.95$. Which statement best interprets the value of $r$ for the relationship between pages read and time spent reading?
17. A student council post claims: "Most students support extending lunch by 10 minutes." The post is based on an online poll link shared on the student council's social media page; 312 students responded out of about 1,800 students at the school, and 68% of respondents clicked "support." Which statement best describes whether the conclusion is justified?
18. A researcher fits a line to predict the number of calories burned from minutes spent cycling: $y = 9x + 80,$ where $x$ is minutes cycling (minutes) and $y$ is predicted calories burned (calories). Which interpretation of the slope is correct? (Include units.)
19. Scores on a large district's standardized math test are roughly symmetric and bell-shaped, so a normal model is appropriate. The scores have mean $\mu=70$ and standard deviation $\sigma=10$. Approximately what percent of students score above 80? (Use the empirical rule.)
20. A card is drawn from a set of 12 equally likely cards labeled $A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4$. Let event $A$ be "the letter is $B$," and let event $B$ be "the number is even." What is $P(A\mid B)$? Write your answer as a simplified fraction.

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