Practice Test 9 - 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 restaurant offers 2 sandwich types (Chicken, Veggie) and 3 sides (Chips, Salad, Fruit). A customer randomly chooses one sandwich type and one side. The equally likely outcomes are: $\{(Chicken,Chips),(Chicken,Salad),(Chicken,Fruit),(Veggie,Chips),(Veggie,Salad),(Veggie,Fruit)\}$ Let event $A$ be "the side is Salad," and let event $B$ be "the sandwich is Veggie." What fraction of outcomes in $B$ also belong to $A$? Write your answer as a simplified fraction.
2. A fair die is rolled and a fair coin is flipped. The 12 outcomes in the sample space are equally likely. Define the random variable $X$ as follows: $X=1$ if the coin shows heads and the die shows a number greater than 4; otherwise $X=0$. Which table correctly represents the theoretical probability distribution of $X$?
3. A factory tests products in two stages. First, a product is randomly selected from the day's output; $P(A)=0.12$ that it is defective. Then, the product goes through a scanner; if the product is defective, the scanner flags it with probability $P(B\mid A)=0.85$. What is the probability that both events occur (the product is defective and then the scanner flags it), i.e., $P(A\cap B)$?
4. A bag contains 6 red marbles and 4 blue marbles. Three marbles are drawn without replacement. Order does not matter because only the set of colors drawn matters. There are $\binom{10}{3}$ equally likely 3-marble selections. What is the probability that exactly 2 of the marbles drawn are red?
5. A game developer compared the number of attempts players needed to beat a level in two versions of the game. Data Set A (Version A): 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7 Data Set B (Version B): 2, 3, 3, 4, 4, 5, 5, 6, 7, 9, 15 Data Set B is right-skewed with a high outlier (15). Which statement best compares the typical number of attempts and the variability using appropriate measures (median and IQR)?
6. A factory produces bolts whose lengths are roughly symmetric and bell-shaped, so a normal model is appropriate. Bolt lengths are modeled as $N(\mu=50\text{ mm},\,\sigma=2\text{ mm})$. Approximately what percent of bolts are between 48 mm and 52 mm? (Use the empirical rule.)
7. A bag contains cards labeled 1, 2, and 5. One card is drawn at random. Let $X$ be the number on the card. The probabilities are $P(X=1)=\tfrac{1}{2}$, $P(X=2)=\tfrac{1}{3}$, and $P(X=5)=\tfrac{1}{6}$ (they sum to 1). What is the expected value of $X$?
8. A school administrator wants to examine the relationship between sleep and grades. For one month, volunteers from the school record how many hours they sleep each night, and the administrator later compares their average sleep hours with their report-card averages. No one is told to change their sleep. Which type of study is described?
9. A fair coin is tossed twice. Let $X$ be defined as the number of tails obtained. Which description correctly defines the random variable $X$?
10. Six students—A, B, C, D, E, and F—are all qualified to lead a lab group, and the teacher wants a fair selection. Fair means each student has probability $1/6$ of being chosen. Which method is fair?
11. A card is drawn at random from 6 equally likely cards labeled $A,B,C,D,E,F$. Define the random variable $X$ by the mapping: $\{A\}\mapsto 0$, $\{B,C\}\mapsto 1$, and $\{D,E,F\}\mapsto 3$. Which table correctly represents the theoretical probability distribution of $X$?
12. A teacher randomly assigned 60 students to try two different study plans for a history quiz: Treatment A (spaced practice) and Treatment B (rereading). Thirty students were randomly assigned to each plan. The mean quiz score was 82.4 for Treatment A and 76.1 for Treatment B, so the observed difference in means was $82.4-76.1=6.3$ 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 study-plan labels 2000 times and recalculating (A − B) each time. In the 2000 shuffles, 18 simulated differences were at least as large as $6.3$. Which conclusion is most reasonable about the treatment effect?
13. A teacher shares a "study tip" slide: "Listening to music while studying improves test scores." The slide summarizes a classroom experiment: 60 students were randomly assigned to study the same review sheet for 20 minutes either with instrumental music (n=30) or in silence (n=30). On the next day's quiz, the music group averaged 84 and the silence group averaged 78. Which statement best describes whether the conclusion is justified?
14. A product quality check defines event $A$ as "an item has a scratch" and event $B$ as "an item has a dent." The inspection data show $P(A)=0.22$, $P(B)=0.15$, and $P(A\cap B)=0.06$. What is the probability that an item has a scratch or a dent (inclusive or)?
15. A streaming service models monthly data usage with $y = 3.2x + 15,$ where $x$ is the number of hours streamed in a month and $y$ is the predicted data used in gigabytes (GB). What does the y-intercept represent in this context? (Include units.)
16. To test whether screen brightness affects battery life, a lab technician randomly assigns 80 identical phones to one of two settings for a standardized video playback test: 30% brightness or 90% brightness. Phones at 90% brightness have shorter average battery life (fewer minutes). Which statement is the most reasonable conclusion?
17. A town wants to estimate the mean number of minutes it takes residents to commute to work. The town randomly selects 80 residents from a list of all residents and asks each person for their commute time. Which statement is correct about the statistic and the parameter?
18. A teacher wants to know whether using a spaced-practice app increases quiz scores. From all students in her classes, she randomly assigns half to use the app for 3 weeks and half to study as they usually do. At the end, everyone takes the same quiz. What does the randomization in this study allow the teacher to conclude?
19. A student collected paired data on the number of practice problems completed (x) and the time to finish a quiz in minutes (y). Technology reports $r=-0.52$. Which statement best interprets the value of $r$?
20. In a randomized experiment, 80 volunteers were randomly assigned to drink either a caffeinated beverage (about 100 mg caffeine) or a caffeine-free beverage each morning for one week. At the end of the week, their average reaction time on a computer test (quantitative) was recorded. The caffeinated group had faster average reaction times than the caffeine-free group. Which statement best describes what the data show?

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