Practice Test 13 - 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 data set's range is much larger than its IQR. What does this suggest?
2. A runner measured reaction times (in seconds) for 11 starts: 0.21, 0.22, 0.22, 0.23, 0.23, 0.24, 0.24, 0.24, 0.25, 0.26, 0.45. Which statement best describes the shape and typical value of the distribution?
3. Quality control context: Event A is "a bottle is underfilled," and event B is "the bottle came from Line B." In one day, 2% of all bottles are underfilled. On Line B, 6% of bottles are underfilled. Which statement best describes whether A and B are independent?
4. A factory makes metal rods whose lengths are approximately normally distributed because many small sources of variation affect the final length. The rod lengths have mean $\mu=50.0$ cm and standard deviation $\sigma=0.5$ cm. Approximately what percent of rods have lengths between 49.5 cm and 50.5 cm? (Use the empirical rule.)
5. A fair coin is flipped twice. The 4 equally likely outcomes are: HH, HT, TH, TT. Event $B$: at least one flip is H. Event $A$: the two flips match (HH or TT). From the outcomes listed, what is $P(A\mid B)$? Give your answer as a simplified fraction.
6. The lengths of metal rods cut by a calibrated machine are approximately normally distributed. Rod lengths have mean $\mu=120$ cm and standard deviation $\sigma=2$ cm. Approximately what percent of rods are above 124 cm? (Use the empirical rule.)
7. Raffle: 1,000 tickets at $2 each; prizes: one $800, three $100. Find expected profit per ticket.
8. A company tested two website designs to see which leads to more newsletter sign-ups. They randomly assigned 80 visitors to Treatment A (new design) and 80 visitors to Treatment B (old design). In Treatment A, 44 of 80 visitors signed up ($\hat p_A=0.55$). In Treatment B, 28 of 80 visitors signed up ($\hat p_B=0.35$). The observed difference in proportions was $0.55-0.35=0.20$ (A − B). Under the assumption of no treatment effect, the analyst shuffled the "A/B" labels 5000 times and computed the difference in sign-up rates each time. In the 5000 shuffles, 6 differences were at least as large as 0.20 (A − B). Based on the randomization test, is the observed difference surprising under no effect?
9. The lifetimes of a certain brand of light bulb are approximately normal (based on quality-control data, the distribution is roughly bell-shaped). Lifetimes have mean $\mu = 1000$ hours and standard deviation $\sigma = 100$ hours. Approximately what percent of bulbs last between 900 and 1100 hours? (Use the empirical rule.)
10. A club tracked how many push-ups 15 members completed in one minute: 14, 16, 18, 18, 19, 20, 20, 20, 21, 22, 22, 23, 24, 26, 28. Which statement correctly describes the distribution shown by a dot plot?
11. A fair six-sided die is rolled once (each of $1$–$6$ equally likely). Define the random variable $X$ by the mapping: odd outcomes $\{1,3,5\}\mapsto 1$ and even outcomes $\{2,4,6\}\mapsto 4$. Which table correctly represents the theoretical probability distribution of $X$?
12. Two fair coins are flipped. The equally likely outcomes are $\{HH, HT, TH, TT\}$. Define the random variable $X$ as the number of heads obtained. Which table correctly represents the theoretical probability distribution of $X$?
13. Sports/game context: Event A is "a soccer player scores on a penalty kick," and event B is "the player listened to a specific warm-up playlist." The team's data show that 75% of all penalty kicks are scored, and 75% of penalty kicks are scored among players who listened to the playlist. Which statement best describes whether A and B are independent?
14. A spinner is spun 60 times with these observed results: outcome 1 occurred 9 times, outcome 2 occurred 21 times, outcome 3 occurred 18 times, and outcome 4 occurred 12 times. Let $X$ be the number landed on. Using the observed relative frequencies as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
15. A school counselor collected data on 12 students relating hours of sleep the night before a test (hours) and test score (points). A scatter plot of the data shows a generally upward trend, and technology reports a correlation coefficient of $r=0.78$. Which statement best interprets the value of $r$ for the relationship between hours of sleep and test score?
16. A store analyzes product attributes. Event $A$ is "an item is on sale," event $B$ is "an item is a clearance item." The store finds $P(A)=0.35$, $P(B)=0.25$, and $P(A\cap B)=0.10$. What is the probability that $A$ or $B$ occurs (inclusive "or")?
17. Three movie options—Comedy, Action, and Documentary—are being considered for a class reward day. The class wants a fair method to select one, meaning each option should have probability $1/3$ of being chosen. Which procedure is fair?
18. A coach randomly assigned 40 basketball players to two free-throw practice routines for one week: Treatment A (short daily sessions) and Treatment B (one long weekly session). Twenty players were randomly assigned to each routine. At the end of the week, the mean number of made free throws out of 20 was 15.1 for Treatment A and 13.4 for Treatment B, so the observed difference in means was $15.1-13.4=1.7$ made shots (A − B). The coach ran a randomization test by shuffling the routine labels 1000 times and recalculating (A − B). In the 1000 shuffles, 6 simulated differences were $1.7$. Which conclusion is most reasonable about the treatment effect?
19. A pet shelter recorded the number of days 14 dogs stayed before being adopted: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 29 How does the outlier (29 days) affect the mean and median number of days?
20. Two clubs, Art and Robotics, are tied for which gets the larger meeting room. They agree the decision should be fair, meaning each club has probability $1/2$ of getting the room. Which procedure is fair?

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