Practice Test 25 - 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 company randomly assigned 50 customer-service trainees to one of two training modules: Treatment A (interactive practice) and Treatment B (video-only). There were 25 trainees per group. After training, each trainee completed a timed task; the outcome was the number of correct entries in 10 minutes. Treatment A mean = 27.4 correct entries, Treatment B mean = 24.9 correct entries, so the observed difference (A − B) is $2.5$. To check if this could happen by chance under "no treatment effect," the company shuffled the treatment labels 5,000 times and computed (A − B) each time. In the shuffled results, 7 of the 5,000 differences were at least as large as $2.5$. Do the results provide evidence that Treatment A is more effective than Treatment B (one-sided, A − B)?
2. A card is drawn at random from a set of 5 cards labeled 1, 1, 2, 3, 3. Define the random variable $X$ as the number on the selected card. Which table correctly represents the probability distribution of $X$?
3. School context: Two events are defined. Event A: a student volunteers at the school fundraiser. Event B: a student is a senior. The student council says, "Of those who volunteered, 40% are seniors." Which statement best describes what "the chance of B given that A" means in this situation?​​
4. The time (in minutes) it takes trained workers to complete a routine assembly task is roughly bell-shaped, so a normal model is reasonable. Completion times have mean $\mu=30$ and standard deviation $\sigma=4$. Approximately what percent of workers finish the task between 26 and 34 minutes? (Use the empirical rule.)
5. In a survey of adults, event $A$ is "a randomly selected person prefers coffee," and event $B$ is "a randomly selected person prefers tea." Suppose $P(A)=0.55$, $P(B)=0.40$, and $P(A\cap B)=0.20$. Using inclusive or (coffee or tea or both), what is $P(A\cup B)$?
6. Two delivery routes were timed (in minutes) over several days. Data Set A is right-skewed with an outlier day due to a traffic accident. Data Set B is roughly symmetric. Summary statistics: - Data Set A: mean $\bar{x}=38.5$, median $=35.0$, standard deviation $s=10.2$, IQR $=5$ - Data Set B: mean $\bar{x}=36.8$, median $=37.0$, standard deviation $s=3.1$, IQR $=4$ Which statement best compares the typical value and variability of Data Set A and Data Set B using appropriate measures?
7. A rideshare company models the predicted cost of a trip with the linear model $y = 2.50x + 4.00,$ where $x$ is the trip distance in miles and $y$ is the predicted cost in dollars. What does the slope represent in this context? (Include units in your interpretation.)
8. A researcher randomly assigned 90 participants to two versions of a logic puzzle app: Treatment A (with hints) and Treatment B (no hints). Forty-five participants were randomly assigned to each version. The proportion who solved the final challenge was 30/45 = 0.667 for Treatment A and 20/45 = 0.444 for Treatment B, so the observed difference in proportions was $0.667-0.444=0.223$ (A − B). Under "no effect," the researcher shuffled the app-version labels 2000 times while keeping the same 50 total solvers. In the 2000 shuffles, 9 simulated differences were $0.223$. Based on the randomization test, is the observed difference surprising under no effect?
9. A calibration error (in units) for a sensor is approximately normal, centered at 0. The mean error is $\mu=0$ and the standard deviation is $\sigma=3$. Approximately what percent of errors are between $-6$ and $6$? (Use the empirical rule.)
10. A tutoring center advertises: "Our program increases quiz scores by 15 points!" Their report compares 20 students' average quiz score the month before tutoring (average 70) to the month after tutoring (average 85). The report does not mention any comparison group of students who did not attend tutoring. Which critique best evaluates the claim?
11. Four players—A, B, C, and D—need a fair way to decide who goes first in a game. Fair means each player has probability $\tfrac{1}{4}$ of going first. Which procedure is fair?
12. In a randomized experiment, 120 volunteers were randomly assigned to drink either a caffeinated beverage or a caffeine-free beverage before taking a 10-minute reaction-time test. The caffeinated group had faster average reaction times (lower times in seconds). Can we conclude that caffeine causes faster reaction times? Why or why not?
13. A survey asks a randomly selected student how many pets they have. Suppose the only possible responses and probabilities are: 0 pets (0.40), 1 pet (0.35), 2 pets (0.20), and 3 pets (0.05). Let $X$ be the number of pets. Which graph represents the distribution of $X$?
14. A set of 12 equally likely outcomes is listed below (each outcome is one card): S1, S2, S3, S4, M1, M2, M3, M4, L1, L2, L3, L4. Event $B$: the letter is S or M. Event $A$: the number is 4. From the outcomes listed, what fraction of outcomes in $B$ also belong to $A$? Give your answer as a simplified fraction.
15. Four classmates—Hana, Idris, Jun, and Lila—need a fair way to decide who will clean up after a science demo. Fair means each classmate has probability $1/4$ of being chosen. Which procedure is fair?
16. At a university, event $A$ is "a randomly selected student is a first-year student," and event $B$ is "a randomly selected student lives on campus." Suppose $P(A)=0.28$, $P(B)=0.62$, and $P(A\cap B)=0.18$. Using inclusive or, what is $P(A\cup B)$?
17. A game app claims its bonus wheel lands on Gold with probability $P(\text{Gold})=0.20$ each spin. A player spins the wheel 40 times and gets Gold 19 times. To check the claim, you simulate 100 runs of 40 spins each using $P(\text{Gold})=0.20$. In the simulations, 0 out of 100 runs produced 19 or more Golds ("as extreme or more extreme" means $\ge 19$ Golds in 40 spins). Which conclusion is most reasonable based on the simulations?
18. A delivery app sampled 100 orders and recorded the tip amount (in dollars). Observed counts: $X=0$ (12 orders), $X=2$ (34 orders), $X=4$ (41 orders), $X=7$ (13 orders). Using the relative frequencies from this sample as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
19. A jar contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. Let event $A$ be "the first marble is red," and event $B$ be "the second marble is blue." Given $P(A)=\frac{5}{8}$ and $P(B\mid A)=\frac{3}{7}$, what is the probability that both events occur, $P(A\cap B)$?
20. Four students—Amina, Ben, Carlos, and Dina—need a fair way to choose who will present first. Fair means each student has the same probability (each $\tfrac14$) of being chosen, even if the outcome isn't "equal" over time. Which procedure is fair?

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