Practice Test 34 - 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 card is chosen at random from the 10 equally likely cards listed below: 1R, 2R, 3R, 4R, 5R, 1B, 2B, 3B, 4B, 5B (Here R = red and B = blue.) Let event $A$ be "the number is greater than 3," and let event $B$ be "the card is red." Using the outcomes listed, what is $P(A\mid B)$? Give your answer as a fraction.
2. In a school survey, event $A$ is "a randomly selected student plays a sport" and event $B$ is "a randomly selected student is in the band." The survey reports $P(A)=0.55$, $P(B)=0.40$, and $P(A\cap B)=0.18$. What is $P(A\cup B)$, where "or" is inclusive (sport or band or both)?
3. A gym models a member's monthly bill with $y = 20x + 35,$ where $x$ is the number of personal training sessions in a month (sessions) and $y$ is the total monthly cost (dollars). What does the y-intercept represent in this context? (Include units.)
4. Using the outcomes listed in $S=\{R1,R2,R3,B1,B2,B3,G1,G2,G3\}$ (one token is chosen; the letter is the color Red/Blue/Green and the number is 1–3), event $A$ is "the token is Blue" and event $B$ is "the number is 2." Which set of outcomes represents $A\cup B$ (A or B)?
5. A website claims that 10% of visitors click an ad, so $P(\text{click})=0.10$ for each visitor. In a sample of 50 visitors, 11 clicked. You simulate 50 samples of 50 visitors under the model. In the 50 simulated samples, 18 samples had 11 or more clicks ("as extreme or more extreme" means $\ge 11$ clicks). What does the simulation suggest about how unusual the observed result is?
6. A dartboard game claims a player hits the bullseye with probability $P(\text{bullseye})=0.20$. In 30 throws, the player hits 2 bullseyes. A student simulates 100 sets of 30 throws under $P(\text{bullseye})=0.20$; 6 of the 100 simulated sets had 2 or fewer bullseyes. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as getting $\le 2$ bullseyes out of 30.)
7. A game has two sequential steps. A player first lands on a bonus square (event $A$). Given that the player landed on the bonus square, the player then draws a "win" card (event $B$). If $P(A)=0.10$ and $P(B\mid A)=0.25$, what is the probability that both $A$ and $B$ occur (i.e., $P(A\cap B)$)?
8. A market research poll measures opinions. Event $A$ is "a respondent supports Proposal X," event $B$ is "a respondent supports Proposal Y." The poll finds $P(A)=0.41$, $P(B)=0.37$, and $P(A\cap B)=0.19$. What is the probability that $A$ or $B$ occurs (inclusive "or")?
9. A game uses one draw from a bag containing 2 green tokens and 2 yellow tokens. One token is drawn at random. Let $X$ be your winnings (in dollars): if you draw green, you win $2; if you draw yellow, you lose $1 (so $X=-1$). Which table correctly represents the probability distribution of $X$?
10. A fair six-sided die is rolled once. All outcomes are equally likely. Define the random variable $X$ by mapping outcomes to values as follows: $\{1,2,3\}\mapsto 0$, $\{4,5\}\mapsto 1$, and $\{6\}\mapsto 3$. Which table correctly represents the theoretical probability distribution of $X$?
11. A simple wager works like this: you pay $3 to play. With probability $0.25$ you receive $12; with probability $0.75$ you receive $0. From the player's perspective, what is the expected value (long-run average net gain/loss) of 1 play?
12. A school district wants to estimate the mean number of minutes of homework completed per night by all 9th-grade students in the district. The district randomly selects 50 9th-grade students from an enrollment list and records each student's minutes of homework last night. Which statement correctly describes the population and the sample?
13. A school recorded whether students prefer watching movies at home or in a theater and whether they have a streaming subscription. The results are shown in the two-way table. What is the conditional relative frequency (as a percent) of students who prefer the theater given that the student does not have a streaming subscription?

    	Prefer Home	Prefer Theater	Total
    Streaming: Yes	58	22	80
    Streaming: No	24	56	80
    Total	82	78	160

14. School context: Event A is "a student is in the morning study hall," and event B is "a student is in the tutoring program." The coordinator says, "Students in tutoring are more likely to attend study hall." Which statement best describes what that claim is comparing?
15. A teacher looked at her class and found that students who spent more minutes per week on math practice websites tended to have higher quiz averages (a positive association). Students chose how much to practice on their own. Which statement is the most reasonable conclusion?
16. A savings account balance grows by a consistent percent each year. The balance (in dollars) is recorded below. Year: 0, 1, 2, 3, 4, 5 Balance ($): 1200, 1260, 1323, 1389, 1458, 1531 Which type of function best models the relationship between year and balance?
17. A researcher found a negative correlation between the number of absences a student had during a semester (quantitative) and the student's final course grade (quantitative): more absences tended to go with lower grades. The researcher used existing attendance and grade records and did not assign absences (observational study). Which statement best describes what the data show?
18. A bag contains 8 equally likely tiles labeled A1, A2, A3, A4, B1, B2, B3, B4. One tile is chosen at random. Event $B$: the tile's letter is B. Event $A$: the tile's number is even. Using the outcomes listed, what is $P(A\mid B)$? Give your answer as a simplified fraction.
19. A coach wants to see whether a new warm-up routine improves 1-mile run times. Two existing PE classes are used: one class does the new warm-up every day for three weeks, and the other class keeps the usual warm-up. The coach flips a coin to decide which class gets the new warm-up. At the end, all students run a timed mile. Which statement is correct about randomization in this study?
20. Scores on a standardized exam are designed to be approximately normal. Suppose scores have mean $\mu=70$ and standard deviation $\sigma=10$. Approximately what percent of students score between 60 and 80? (Use the empirical rule.)

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