Practice Test 40 - 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 game designer randomly assigned 90 players to try either Treatment A (a new tutorial) or Treatment B (the old tutorial), 45 players per group. The outcome was whether the player completed the first level without hints. Results: Treatment A had 33/45 successes ($73.3\%$), Treatment B had 21/45 successes ($46.7\%$). Observed difference in proportions (A − B) = $0.733-0.467=0.266$. A randomization test shuffled the success/failure outcomes across groups 5,000 times under "no treatment effect." In the simulation, 18 of the 5,000 shuffled differences were at least as large as $0.266$. Based on the randomization test, is the observed difference surprising under no effect?
2. A fitness app uses the model $y = 180 - 6x,$ where $x$ is the number of minutes after a workout ends (minutes) and $y$ is a person's heart rate (beats per minute). Which interpretation of the slope is correct? (Include units.)
3. A quality-control measurement is approximately normal. The measurement has mean $\mu=200$ units and standard deviation $\sigma=5$ units. Approximately what percent of measurements are below 190 units? (Use the empirical rule.)
4. A gym wants to estimate the mean number of visits per month for all current gym members. The gym randomly selects 80 members from its membership database and finds the sample mean is 6.2 visits per month. What is a reasonable inference based on the sample?
5. A café tracked the number of customers per hour for 10 hours on a weekday: 18, 19, 20, 20, 21, 21, 22, 22, 23, 45 Which conclusion about the data is best supported?
6. A class recorded how many minutes 12 students spent reading last night: 10, 12, 15, 15, 18, 20, 20, 22, 25, 25, 28, 30. Which statement correctly describes the distribution shown by a box plot of these data?
7. A fitness app recorded the time (in minutes) it took two groups to complete the same 2-mile walk. Data Set A (Group A times): 22, 23, 24, 24, 25, 25, 26, 26, 27, 28, 60 Data Set B (Group B times): 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 30 Data Set A has a clear high outlier (60 minutes), while Data Set B does not. Which statement best compares the typical value and variability using appropriate measures (median and IQR)?
8. A company wants to estimate the proportion of all light bulbs produced this week that are defective. From the week's production, a quality-control technician uses a random-number generator to select 200 bulbs to test; 6 are defective. Why is random sampling important in this situation?
9. At a university, event $A$ is "a student is majoring in Biology" and event $B$ is "a student is taking Calculus this term." The probabilities are $P(A)=0.28$, $P(B)=0.50$, and $P(A\cap B)=0.12$. What is the probability that $A$ or $B$ occurs (inclusive or)?
10. A game awards $X$ points with probabilities: $P(X=0)=\tfrac{1}{4}$, $P(X=4)=\tfrac{1}{2}$, and $P(X=8)=\tfrac{1}{4}$. What is the expected value of $X$? Give your answer in points.
11. A student modeled the relationship between the number of practice problems completed (x) and a quiz score (y) with the linear model $\hat{y}=45+3x$. The residuals (defined as $\text{residual}=y-\hat{y}$) for several students are shown below. x: 1, 2, 3, 4, 5, 6, 7 residual: -6, -4, -2, 0, 2, 4, 6 Which statement best describes how well the model fits the data?
12. Three fair coins are flipped. The $8$ outcomes are equally likely. Define the random variable $X$ as the number of tails obtained. Which table correctly represents the theoretical probability distribution of $X$?
13. 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)." You are given $P(A)=\tfrac{13}{52}$, $P(B)=\tfrac{12}{52}$, and $P(A\cap B)=\tfrac{3}{52}$. What is $P(A\cup B)$, where or is inclusive?
14. A coach wants to know whether athletes who drink water during practice report less fatigue afterward. The coach does not change anyone's routine. After practice, the coach asks each athlete how many minutes they spent drinking water and to rate fatigue on a 1–10 scale. What does the lack of random assignment imply?
15. A two-stage experiment is performed. First flip a fair coin (H or T), then roll a fair number cube (1–6). The 12 equally likely outcomes are: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. Event $B$: the coin shows H. Event $A$: the number rolled is greater than 4. Using the outcomes listed, what is $P(A\mid B)$? Give your answer as a simplified fraction.
16. A principal wants to estimate the average number of hours students spend on homework per week at her school. She stands in the cafeteria during lunch and asks the first 60 students she sees to report how many hours of homework they did last week. Which type of study is described, and what is a key limitation related to randomization?
17. A random sample of 200 students was asked whether they prefer working in groups or alone and whether they prefer math or English. Using the two-way table, estimate the probability that a randomly selected student prefers math given that the student prefers working in groups. Give your answer as a percent.
18. A basketball player claims to make free throws with probability $P(\text{make})=0.70$. In one practice session, the player takes 20 free throws and makes 8. You simulate 50 sessions of 20 shots each under the model ($P=0.70$). In the 50 simulated sessions, 1 session had 8 or fewer makes ("as extreme or more extreme" means $\le 8$ makes). Which conclusion is most reasonable based on the simulations?
19. A quality-control report defines event $A$ as "an item fails the seal test" and event $B$ as "an item fails the barcode scan." Given that $P(A)=0.10$, $P(B)=0.25$, and $P(A\cap B)=\tfrac{1}{40}$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
20. Using the outcomes listed in $S=\{\text{A},\text{B},\text{C},\text{D},\text{E},\text{F}\}$, one letter is selected at random. Event $A$ is the event that the letter is a vowel. Event $B=\{\text{D},\text{E},\text{F}\}$. Which expression represents $A\cup B$ (A or B)?

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