Practice Test 39 - Statistics and Probability

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

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1. In a sample of 80 households, 32 have a pet (event $B$). Of those, 12 also subscribe to a streaming service (event $A \cap B$), where event $A$ is "a household subscribes to a streaming service." Based on the information given, what is $P(A \mid B)$? Give your answer as a simplified fraction.
2. A fair coin is flipped 3 times. The sample space consists of all equally likely length-3 sequences of H and T. Define the random variable $X$ as the number of heads in the 3 flips. Which table correctly represents the theoretical probability distribution of $X$ (using $P(X=x)$ derived from the fraction of outcomes that produce each $x$)?
3. Based on the sample space shown, $S=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)\}$ for rolling a 2-sided die (1–2) and a 3-sided die (1–3) and recording an ordered pair (first, second), let event $A$ be "the sum is 4" and event $B$ be "the first number is 2." Which set of outcomes represents $A\cap B$ (A and B)?
4. A game payout $X$ (in dollars) has distribution: $P(X=2)=\tfrac{1}{2}$, $P(X=4)=\tfrac{1}{4}$, $P(X=10)=\tfrac{1}{4}$. What is the expected value of $X$? Give your answer in dollars.
5. A grocery store recorded the number of items in 10 customers' baskets: 6, 7, 7, 8, 8, 9, 9, 10, 10, 27. Which statement best describes how the outlier affects the mean and median number of items?
6. A teacher claims a multiple-choice quiz with 4 options per question is being guessed randomly, so $P(\text{correct})=0.25$ on each question. A student answers 40 questions and gets 19 correct. A simulation of 150 random-guessing students (40 questions each) shows that 0 out of 150 got 19 or more correct. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as $\ge 19$ correct out of 40.)
7. In a school survey, two events are recorded for a randomly selected student: Event $A$ = "the student participates in a club," and event $B$ = "the student uses the bus to get to school." Given that $P(A)=0.30$, $P(B)=0.40$, and $P(A\cap B)=0.15$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
8. A teacher recorded whether students turned in an assignment on time and whether they used a planner. The results are shown in the two-way table. Which statement best describes the association between using a planner and turning in the assignment on time?

   	On Time	Late	Total
   Planner: Yes	72	18	90
   Planner: No	36	54	90
   Total	108	72	180

9. A spinner has 8 equally likely outcomes labeled 1 through 8. The outcomes are grouped below: - Odd outcomes: 1, 3, 5, 7 - Even outcomes: 2, 4, 6, 8 - Prime outcomes: 2, 3, 5, 7 Let event $A$ be "the outcome is prime," and let event $B$ be "the outcome is odd." Based on the outcomes listed, what fraction of outcomes in $B$ also belong to $A$? Give your answer as a fraction.
10. Adult heights in a certain population are roughly symmetric and bell-shaped, so using a normal model is reasonable. Heights are modeled as $N(\mu=170\text{ cm},\,\sigma=6\text{ cm})$. Approximately what percent of adults are between $164$ cm and $176$ cm? (Use the empirical rule.)
11. From a standard 52-card deck, event $A$ is "the card is red," and event $B$ is "the card is a king." You know $P(A)=\frac{26}{52}$, $P(B)=\frac{4}{52}$, and $P(A\cap B)=\frac{2}{52}$. What is the probability that $A$ or $B$ occurs, where "or" is inclusive (red or king or both)?
12. A club has four project ideas—Garden, Mural, Recycling, and Tutoring—and must pick one to start first. They want a fair random choice, meaning each idea has probability $1/4$ of being selected. Which method gives each idea an equal chance?
13. A teacher randomly assigned 60 students to try two study plans for a vocabulary quiz: Treatment A (spaced practice) and Treatment B (single long review). Thirty students were randomly assigned to each treatment. The outcome was the quiz score (out of 100). Treatment A had a mean score of 84.2 and Treatment B had a mean score of 78.5, for an observed difference in means of $84.2-78.5=5.7$ 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 2000 times and recalculating the difference in means each time. In the simulations, 18 of the 2000 shuffled differences were at least as large as $5.7$ (A − B). Which conclusion is most reasonable about the treatment effect?
14. A bag contains 5 red marbles and 3 blue marbles. You draw one marble first and do not replace it. Let event $A$ be "the first marble is red," so $P(A)=\frac{5}{8}$. Let event $B$ be "the second marble is blue." Given that the first marble was red, $P(B\mid A)=\frac{3}{7}$. What is the probability that both events occur, $P(A\cap B)$?
15. A school club sells a lottery-style ticket for $2. Exactly one of the following outcomes happens when you buy 1 ticket: with probability $0.10$ you win $10; with probability $0.20$ you win $3; with probability $0.70$ you win $0. From the player's perspective, what is the expected value (long-run average net gain/loss) of buying 1 ticket?
16. A card is drawn at random from a set of 3 cards labeled A, B, and C. The sample space is $\{A,B,C\}$ with each outcome equally likely. Define the random variable $X$ as follows: $X(A)=0$, $X(B)=0$, and $X(C)=1$. Which table correctly represents the probability distribution of $X$?
17. A city library wants to estimate the proportion of all adult library cardholders in the city who prefer e-books over printed books. The library has a complete list of adult cardholders and uses a random number generator to select 120 cardholders to survey. In the sample, 48 say they prefer e-books. What population parameter is being estimated using this random sample?
18. A school newspaper reports a link between students' participation in after-school clubs and their GPA. A reporter collects data by looking up GPA in school records and recording each student's number of clubs from the activities database. No one is told to join or not join clubs. Which type of study is described?
19. A local bakery tracks the outside temperature (°F) and the number of iced coffees sold that day. A scatter plot shows a roughly linear upward trend, and technology reports $r=0.91$. Which statement best interprets the value of $r$ for the relationship between temperature and iced coffee sales?
20. A snack company wants to estimate the population parameter $\mu$, the mean rating (out of 10) that all customers would give a new flavor. A random sample of $n=60$ customers produced a sample statistic $\bar{x}=7.4$. Students simulated 5,000 repeated random samples of size 60 and recorded the sample mean each time. In the simulation, the middle 95% of sample means ran from 7.0 to 7.8. What margin of error is supported by the simulation for estimating $\mu$?

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