Practice Test 6 - Statistics and Probability

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

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1. According to $S=\{10,11,12,13,14,15\}$, one number is chosen at random. Event $A$ is the event that the number is a multiple of 3. Which set represents $A^c$ (the complement of $A$, or not $A$)?
2. A researcher tested a small robot 120 times and recorded whether it completed a task successfully. The observed outcomes were: Success (78 trials) and Failure (42 trials). The robot earns 4 points for Success and loses 1 point for Failure. Let $X$ be the points earned on a trial. Using the relative frequencies from the 120 trials as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
3. A study on concentration randomly assigned 90 students to either study in a quiet room (low noise level in decibels) or a room with background music (higher noise level in decibels) for one hour. Students in the quiet room completed more practice problems on average. Can we conclude that lower noise level causes students to complete more practice problems? Why or why not?
4. A warehouse does a two-stage inspection. First, a package is selected; $P(A)=0.30$ that it has a labeling error. Then, given it has a labeling error, the scanner detects the error with probability $P(B\mid A)=0.60$. What is the probability that both the package has a labeling error and the scanner detects it (i.e., $P(A\cap B)$)?
5. From 7 candidates, a committee of 3 is selected at random, and order does not matter. What is the probability that a particular candidate, Jordan, is on the committee?
6. A researcher randomly assigned 80 plants to receive either 0 mL, 10 mL, or 20 mL of fertilizer per week. After 5 weeks, plants that were assigned more fertilizer had greater average height. Which statement best describes what the data show?
7. A student measured the lengths (in cm) of 13 leaves: 6.2, 6.4, 6.5, 6.5, 6.6, 6.7, 6.8, 6.8, 6.9, 7.0, 7.1, 7.3, 7.6. Which plot best represents the data as a box plot summary (median and range only)?
8. A school wants to estimate the proportion of all students at the school who bring lunch from home on a typical day. The office randomly selects 80 student IDs from the full student roster and asks those students whether they brought lunch from home today; 52 say yes. Why is random sampling important in this situation?
9. Three teams—Lions, Tigers, and Bears—are tied and need a fair tiebreaker to decide who gets home-field advantage. Fair means each team has probability $1/3$ of being chosen. Which method gives each team an equal chance?
10. A fair coin is flipped and then a fair six-sided die is rolled. The $12$ outcomes are equally likely. Define the random variable $X$ as $X=1$ if the coin shows heads and the die shows an even number, and $X=0$ otherwise. Which table correctly represents the theoretical probability distribution of $X$?
11. A news article reports: "People who spend more minutes per day on a language-learning app tend to score higher on a vocabulary quiz." The article's data came from app users who chose how much to study; no one was assigned a study time. Which statement best describes what the data show?
12. Scores on a standardized quiz are approximately normal (roughly symmetric and bell-shaped). The distribution is modeled as $N(\mu=70,\,\sigma=10)$. Approximately what percent of students score below $50$? (Use the empirical rule.)
13. In a group of 200 employees, 80 work remotely (event $B$), and 28 both work remotely and are in management (event $A \cap B$), where event $A$ is "the employee is in management." Based on the information given, what is $P(A \mid B)$? Give your answer as a fraction in simplest form.
14. A streaming service reports $P(A)=0.40$, $P(B)=0.25$, and $P(A\cap B)=0.10$, where event $A$ is "a user watches a documentary this week" and event $B$ is "a user watches a comedy special this week." Based on the information given, are $A$ and $B$ independent? Use conditional probability by comparing $P(A\mid B)$ and $P(A)$.
15. Quality control context: Event A is "a part passes the final inspection," and event B is "the part was made on Machine 2." A supervisor says, "Of those made on Machine 2, 95 out of 100 pass final inspection." Which statement best describes what "the chance of A given B" means here?
16. A student newspaper claims: "The new tutoring center helps students pass math." It reports that among 40 students who visited the tutoring center at least twice, 34 passed math this semester. The article does not report pass rates for students who did not use the tutoring center. What information is most needed to judge this report?
17. A game is played by rolling a fair die once. You win $5$ if the roll is 1 or 2, win $1$ if the roll is 3, 4, or 5, and lose $2$ if the roll is 6. Let $X$ be your winnings (in dollars). Which table correctly represents the probability distribution of $X$?
18. A bookstore records $P(A)=0.30$, $P(B)=0.50$, and $P(A\cap B)=0.12$, where event $A$ is "a customer buys fiction" and event $B$ is "a customer buys a bookmark." Based on the information given, are $A$ and $B$ independent? Use conditional probability by comparing $P(A\mid B)$ and $P(A)$.
19. A fair six-sided die is rolled once (outcomes $1$–$6$ equally likely). Define the random variable $X$ by the explicit mapping: $X=1$ if the outcome is $1$ or $2$; $X=2$ if the outcome is $3$; $X=3$ if the outcome is $4$ or $5$; and $X=6$ if the outcome is $6$. Which table correctly represents the theoretical probability distribution of $X$?
20. A student recorded the number of pages read each day for 14 days: 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 30. Which measure of spread is most affected by the outlier value of 30?

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