Practice Test 43 - 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 die is rolled once. Let $A$ be "the result is 1" and $B$ be "the result is 2." Given that $P(A)=\tfrac{1}{6}$, $P(B)=\tfrac{1}{6}$, and $P(A\cap B)=0$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
2. Two cafeterias tracked the number of students arriving during a 10-minute interval each day. Data Set A: 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37 Data Set B: 20, 22, 25, 28, 30, 32, 34, 36, 38, 40, 42, 45 Both distributions are roughly symmetric with no clear outliers. Which statement best compares the typical number of students and the variability using appropriate measures (mean and standard deviation)?
3. A bag contains marbles, and you draw one marble, replace it, then draw again. Let event $A$ be "the first draw is blue," and event $B$ be "the second draw is blue." Given that $P(A)=0.30$, $P(B)=0.30$, and $P(A\cap B)=0.09$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
4. A gym recorded how many minutes members spent on a treadmill during a certain hour. The dot plot shows the times. Dot plot (minutes): 20: • 22: •• 24: ••• 26: ••• 28: •• 30: • 32: • 60: • Which statement best describes the shape and spread of the distribution?
5. A car's stopping distance ($y$, in feet) was measured at different speeds ($x$, in mph). Over this limited range, the relationship appears approximately linear. Which equation best models the relationship between speed and stopping distance?
6. Based on the sample space shown for choosing one ordered pair, $S=\{(1,1),(1,2),(2,1),(2,2)\}$, let event $A$ be "the sum of the two numbers is 3," and let event $B$ be "the first number is 2." Which expression represents $A\cup B$ (A or B)?
7. School context: Event A is "a student turned in the homework on time," and event B is "a student attended the review session." You hear: "Of those who attended the review session, 80% turned in the homework on time." Which statement best describes what "the chance of A given B" means in this situation?
8. An engineer models the output of a machine (y, in units) from the number of hours it has been running since maintenance (x) using the linear model $\hat{y}=100-2x$. Residuals are defined as $\text{residual}=y-\hat{y}$. What does the residual pattern suggest about the model choice?
9. A library wants to estimate the population parameter $p$, the proportion of all library visitors who check out at least one book. A random sample of $n=60$ visitors was observed, and the sample statistic was $\hat p=0.55$ (55%). A simulation repeatedly took 1,000 random samples of size 60 from a large list of library visits and recorded $\hat p$ each time. The simulation summary said: "About 95% of the simulated $\hat p$ values were within 0.08 of 0.55." Which interval is a reasonable estimate range for the population parameter $p$? (Write endpoints as decimals.)
10. A ball is rolled up a ramp and then comes back down. Its distance from the start is measured at equal time intervals. The data are shown. Which type of function is most reasonable for modeling the relationship between time and distance? Time (s): 0, 1, 2, 3, 4, 5 Distance (m): 0, 3, 5, 6, 5, 3
11. A school counselor compared students' number of extracurricular activities (count) with their weekly hours of homework (hours). Technology reported a correlation coefficient of $r = -0.28$. Which statement best interprets the value of $r$ for the relationship between number of activities and homework hours?
12. A bag contains 4 red marbles and 3 blue marbles. Three marbles are drawn without replacement. Order does not matter because only the set of marbles drawn is considered. What is the probability of drawing exactly 2 red marbles and 1 blue marble?
13. A farmer models the total weight of apples harvested with $y = 1.2x + 5,$ where $x$ is the number of apple trees harvested (trees) and $y$ is the total weight harvested (kilograms). What does the $y$-intercept represent in this context? (Include units.)
14. A college health center found that students who reported more cups of water per day tended to report fewer headaches per month (a negative correlation). The data came from a voluntary questionnaire; students were not assigned water intake amounts. Which statement best describes what the data show?
15. A school randomly assigned 60 students to try two different study plans for a vocabulary quiz: Treatment A (spaced practice) and Treatment B (single long review). Each group had 30 students. The quiz was scored out of 50 points. Treatment A had a mean score of 41.2 and Treatment B had a mean score of 38.6, for an observed difference (A − B) of $2.6$ points. To test whether this difference could be due to chance under "no treatment effect," the teacher performed a randomization test by shuffling the treatment labels 1,000 times and recalculating (A − B) each time. In the simulations, 12 of the 1,000 shuffled differences were at least as large as $2.6$. Which conclusion is most reasonable about the treatment effect (using the one-sided direction A − B)?
16. The fill weights of cereal boxes from a machine are checked and the distribution is approximately bell-shaped, so it is reasonable to use a normal model. Assume fill weights are normally distributed with mean $\mu = 368\text{ g}$ and standard deviation $\sigma = 4\text{ g}$. Approximately what percent of boxes have fill weight below $360\text{ g}$?
17. Scores on a standardized math assessment are modeled well by a normal distribution (approximately symmetric and bell-shaped). The scores have mean $\mu = 70$ and standard deviation $\sigma = 10$. Approximately what percent of students score between 60 and 80? (Use the empirical rule.)
18. A company reviews device capabilities. Event $A$ is "a device supports 5G," event $B$ is "a device has at least 256GB of storage." The review shows $P(A)=0.58$, $P(B)=0.36$, and $P(A\cap B)=0.22$. What is the probability that $A$ or $B$ occurs (inclusive "or")?
19. In a randomized experiment, 48 students were randomly assigned to two different methods for learning keyboard shortcuts: Treatment A (game-based practice) and Treatment B (worksheet practice). There were 24 students per group. After one week, the outcome was the number of shortcuts correctly recalled (out of 30). Treatment A mean = 21.5, Treatment B mean = 18.9, so the observed difference (A − B) is $2.6$. A randomization test was conducted by shuffling the 48 recall scores between groups 2,000 times under "no treatment effect." In the simulations, 28 of the 2,000 shuffled differences were at least as large as $2.6$. Which conclusion is most reasonable about the treatment effect (one-sided, A − B)?
20. A factory produces light bulbs. A bulb is selected at random and first tested for being defective (event $A$). If it is defective, it is then checked to see whether the defect is visible (event $B$). If $P(A)=0.06$ and $P(B\mid A)=0.70$, what is the probability that both $A$ and $B$ occur (i.e., $P(A\cap B)$)?

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