Practice Test 41 - Statistics and Probability

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

A bag contains 3 red marbles (R) and 2 blue marbles (B). One marble is drawn at random; each marble is equally likely to be drawn. Define the random variable $X$ by $X=2$ if a red marble is drawn and $X=-1$ if a blue marble is drawn. What is the expected value of $X$ based on this theoretical model?
A student blog posts: "The new morning announcement format reduced tardiness by 50%!" The blog reports that on one Monday there were 20 tardies, and on the next Monday (after the new format started) there were 10 tardies. No other days are mentioned. Which critique best evaluates the claim?
A company models the total cost to produce custom T-shirts with the linear model $y = 4.5x + 120,$ where $x$ is the number of T-shirts (shirts) and $y$ is the total cost (dollars). What does the slope represent in this context? (Include units.)
A trivia app claims users answer each question correctly with probability $P(\text{correct})=0.50$ (random guessing). A user answers 10 questions and gets 9 correct. You simulate 100 runs of 10 questions under the guessing model. In the 100 simulated runs, 1 run had 9 or more correct ("as extreme or more extreme" means $\ge 9$ correct). What does the simulation suggest about how unusual the observed result is?
Based on the sample space $S=\{\text{A},\text{B},\text{C},\text{D},\text{E}\}$ for randomly choosing one letter tile, event $A$ is "the letter is a vowel" and event $B$ is "the letter comes after C in the alphabet (within S)." Which set represents $A\cap B$ (A and B)?
A card is drawn at random from a standard 52-card deck. Event $A$ is "the card is a heart," and event $B$ is "the card is a face card (J, Q, or K)." Based on the information given that $P(A\cap B)=\frac{3}{52}$ and $P(B)=\frac{12}{52}$ , what is $P(A\mid B)$ ? Give your answer as a simplified fraction.
A company's package fill weights are approximately normal based on a stable process (roughly symmetric and bell-shaped). Fill weights are modeled as $N(\mu=500\text{ g},\,\sigma=20\text{ g})$ . Approximately what percent of packages are under $460$ g? (Use the empirical rule.)
A school models the number of pages a printer can produce as $y = 30x + 200,$ where $x$ is the number of hours the printer has been running (hours) and $y$ is the total number of pages printed (pages). What does the slope represent in this context? (Include units.)
A video game claims that a rare item drops with probability $P(\text{drop})=0.15$ each time a player defeats a boss. A player defeats the boss 30 times and gets 2 drops. A student simulates 100 sets of 30 defeats under $P(\text{drop})=0.15$ ; 18 of the 100 simulated sets had 2 or fewer drops. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as getting $\le 2$ drops in 30 tries.)
A cafeteria randomly assigned 40 students to receive either Treatment A (fruit placed at eye level) or Treatment B (fruit placed on a lower shelf), 20 students per group. The outcome was whether the student chose fruit. Results: Treatment A fruit choice = 14/20, Treatment B fruit choice = 6/20. Observed difference in proportions (A − B) = $0.70-0.30=0.40$ . A randomization test shuffled the 40 yes/no outcomes across the two groups 1,000 times under "no treatment effect," recalculating (A − B) each time. In the simulation, 3 of the 1,000 shuffled differences were at least as large as $0.40$ . Which conclusion is most reasonable about the treatment effect?
A club has 8 students: 5 seniors and 3 juniors. The club randomly selects a 3-person committee. Order does not matter because committee roles are not assigned. What is the probability that the committee has exactly 2 seniors and 1 junior?
A survey company claims that $P(\text{supports the proposal})=0.40$ for the population. They survey 80 people and find 28 support the proposal. You simulate 100 samples of 80 people each using $P=0.40$ . In the simulations, 41 out of 100 samples had 28 or fewer supporters ("as extreme or more extreme" means $\le 28$ ). Which conclusion is most reasonable?
A restaurant tracked 16 delivery times (in minutes) for a busy Friday night: 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 15, 38 How does the outlier (38 minutes) affect the mean and median delivery time?
A school reports that $P(A\cap B)=0.06$ and $P(B)=0.20$ , where event $A$ is "a randomly selected student is in band" and event $B$ is "a randomly selected student plays a sport." Based on the information given, which statement correctly interprets $P(A\mid B)$ ?
A store is deciding which coupon to offer customers. Consider the store's average discount cost per customer (in dollars). Strategy A: Give every customer a guaranteed $3 discount. Strategy B: Run a scratch-off: with probability 0.10 the customer gets $30 off; with probability 0.90 the customer gets $0 off. Over many repetitions, which strategy has the greater expected value of discount cost to the store (higher average discount in dollars)?
A community center randomly assigned 72 participants to two ways of learning a new card game: Treatment A (in-person demonstration) and Treatment B (instruction booklet). There were 36 participants in each group. The outcome was whether the participant could correctly play a full round without help (success/failure). In Treatment A, 27 of 36 succeeded ( $\hat p_A=0.75$ ). In Treatment B, 18 of 36 succeeded ( $\hat p_B=0.50$ ). The observed difference in proportions was $0.75-0.50=0.25$ (A − B). A randomization test shuffled the success/failure outcomes across the two groups 4000 times under "no treatment effect." In the simulations, 40 of the 4000 shuffled differences were at least as large as 0.25 (A − B). Which conclusion is most reasonable about the treatment effect?
A random sample of 180 students was asked whether they take an art class and whether they attend at least one school event (game, concert, etc.) each month. Using the two-way table, are the events A = "takes an art class" and B = "attends at least one school event each month" independent? Use the table by comparing $P(A\mid B)$ to $P(A)$ . Two-way table (counts): - Rows: Takes art class (Yes, No) - Columns: Attends monthly event (Yes, No) Art Yes: Event Yes 50, Event No 10 (Row total 60) Art No: Event Yes 60, Event No 60 (Row total 120) Column totals: Event Yes 110, Event No 70, Grand total 180
Two books—Book X and Book Y—are equally popular, and the librarian needs a fair tie-breaker to decide which one gets displayed first. Fair means each book has probability $1/2$ of being chosen. Which procedure is fair?
A school newsletter reports: "Students who use the library after school get higher grades." The newsletter cites a survey of 40 students who were already in the library after school on one Tuesday. Of those students, 30 reported having an A or B average, and 10 reported a C average or lower. The newsletter concludes that using the library after school causes better grades for students at the school. Which critique best evaluates the claim?
A teacher uses the model $\hat{y}=5x+50$ to predict a student's exam score (y) from the number of hours studied (x). Residuals are defined as $\text{residual}=y-\hat{y}$ . For a student who studied 6 hours, the residual is $-4$ . What does this residual mean in context?

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