Practice Test 11 - 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 sports drink ad says: "Drinking HydrateNow makes runners faster." In a study, 30 runners were randomly assigned to drink HydrateNow or plain water before a 1-mile run. The HydrateNow group averaged 7:50, and the water group averaged 8:05. Which critique best evaluates the claim?
2. A company tested a new typing course by randomly assigning 200 employees to either take the course or not take it. After four weeks, the course group had a higher average typing speed (words per minute). Which statement is the most reasonable conclusion?
3. A teacher says: "Our class's average quiz score is 90%." The teacher explains that 18 out of 20 students scored at least 90% on the quiz. Which critique best evaluates the claim?
4. Researchers randomly assigned 120 students to either listen to music while studying or study in silence for 30 minutes. Afterward, each student took the same vocabulary quiz and received a score (quantitative). The music group had a lower average quiz score than the silence group. Which statement is the most reasonable conclusion?
5. At a school, two events are defined: Event A = "a student arrives late to first period," and Event B = "a student takes the bus to school." The office reports: among bus riders, 12% arrive late; among non-bus riders, 12% arrive late. Which statement best describes whether A and B are independent?
6. A factory inspects products in two steps. First, a product is tested for electrical safety (event $A$). If it passes, it is then checked for cosmetic defects (event $B$). The probability a product passes the electrical safety test is $P(A)=0.80$, and the probability it passes the cosmetic check given it passed safety is $P(B\mid A)=0.90$. What is the probability that both $A$ and $B$ occur (i.e., the product passes both tests)?
7. A student makes a simple wager: pay $4 to play. With probability $0.30$ you win $12; otherwise you win $0. From the player's perspective, what is the expected value (long-run average net gain/loss) of 1 play?
8. Two fair coins are flipped. The equally likely outcomes are listed below: $\{HH,\ HT,\ TH,\ TT\}$ Let event $A$ be "exactly one head," and let event $B$ be "the first coin is heads." What fraction of outcomes in $B$ also belong to $A$? Write your answer as a simplified fraction.
9. A school principal wants to estimate the proportion of all students at the school who bring a packed lunch on a typical day. The principal randomly selects 80 students from the school roster and asks whether they brought a packed lunch today; 36 say yes. Why is random sampling important in this situation?
10. A fitness app analyzed its users and found a negative correlation between daily minutes of exercise and resting heart rate: users who exercised more tended to have lower resting heart rates. The app only used existing user data and did not assign exercise levels. Which statement best describes what the data show?
11. A call center finds that call-handling times are approximately normal for a certain type of call. Assume handling times are normally distributed with mean $\mu=6$ minutes and standard deviation $\sigma=1.5$ minutes. Approximately what percent of calls take below 4.5 minutes? (Use the empirical rule.)
12. A video game level has two sequential requirements. First, the player must find a key (event $A$), which happens with probability $P(A)=0.60$. Then, given the key was found, the player opens the door successfully (event $B$) with probability $P(B\mid A)=0.75$. What is the probability that both events occur, $P(A\cap B)$?
13. A student pays $2 to draw one card from a small deck of 10 cards: 3 cards say "Win $4," 2 cards say "Win $1," and 5 cards say "Win $0." These are the only outcomes. What is the expected value (long-run average net gain/loss) for the player for one draw, including the $2 cost?
14. A deck of 52 cards is shuffled and one card is drawn. Let event $A$ be drawing a heart, and event $B$ be drawing a face card (J, Q, or K). Given $P(A)=\frac{13}{52}$, $P(B)=\frac{12}{52}$, and $P(A\cap B)=\frac{3}{52}$, what is $P(A\cup B)$ (inclusive or: heart, face card, or both)?
15. Normal data have mean 500 and SD 100. What score is the 90th percentile?
16. A counselor recorded the number of colleges each of 20 seniors applied to: 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8. Which statement correctly describes the distribution when shown in a box plot?
17. A student council wants to estimate the proportion of students at their high school who would support adding a 10-minute break between 2nd and 3rd period. They use a random number generator to select 120 students from the school enrollment list and email them a short questionnaire asking whether they support the change. Which type of study is described?
18. A community center randomly assigned 50 volunteers to two different reminder methods for showing up on time: Treatment A (text reminder) and Treatment B (email reminder). Twenty-five volunteers were randomly assigned to each method. On-time arrival occurred for 21 of 25 in Treatment A and 14 of 25 in Treatment B. The observed difference in proportions was $0.84-0.56=0.28$ (A − B). To assess whether this could be due to chance under "no difference," the center performed a randomization test by shuffling the reminder labels 5000 times while keeping the same 35 total on-time arrivals. In the 5000 shuffles, 41 simulated differences in proportions were $0.28$. Do the results provide evidence that Treatment A is more effective than Treatment B?
19. A city wants to estimate the population parameter $\mu$, the mean one-way commute time (in minutes) for all working adults in the city. A random sample of 50 adults had a sample mean (sample statistic) of $\bar x = 34.2$ minutes. Students used repeated-sampling simulation: they generated 1,000 random samples of size 50 (modeling the same sampling process) and recorded the sample mean each time. In the simulation, about 95% of the simulated sample means were within 3.5 minutes of 34.2. What margin of error is supported by the simulation for estimating the population parameter $\mu$?
20. A company measures the diameters of ball bearings it produces. The diameters are roughly symmetric and bell-shaped, so a normal model is appropriate. Assume diameters are normally distributed with mean $\mu = 10.00\text{ mm}$ and standard deviation $\sigma = 0.02\text{ mm}$. Approximately what percent of ball bearings have a diameter between $9.96\text{ mm}$ and $10.04\text{ mm}$?

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