Practice Test 21 - Statistics and Probability

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

A club runs a two-step registration process. First, an applicant submits an online form (event $A$ ). Given that the form is submitted, the applicant then pays the membership fee within 24 hours (event $B$ ). If $P(A)=0.70$ and $P(B\mid A)=0.60$ , what is the probability that both $A$ and $B$ occur, $P(A\cap B)$ ?
A die is claimed to be fair, so $P(6)=\frac{1}{6}$ on each roll. You roll it 30 times and get twelve 6s. You simulate 40 runs of 30 rolls using a fair die. In the 40 simulated runs, 0 runs had 12 or more 6s ("as extreme or more extreme" means $\ge 12$ sixes). Would this result cause you to question the model? Why?
A city wants to estimate the population parameter $\mu$ , the mean number of minutes it takes residents to commute to work. A random sample of $n=50$ residents reported their commute times, and the sample statistic was $\bar x=32$ minutes. A simulation repeatedly took 2,000 random samples of size 50 from a large database of city commutes and recorded $\bar x$ each time. In the simulation, about 95% of the simulated sample means were between 28 and 36 minutes. What margin of error is supported by the simulation for estimating $\mu$ ?
A company models the value of a laptop with $y = -120x + 1500,$ where $x$ is the age of the laptop (years) and $y$ is its resale value (dollars). What does the slope represent in this context? Include units.
From a set of inspection records, event $A$ is "a randomly selected item passes the visual check," and event $B$ is "the same item passes the weight check." Given that $P(A)=0.80$ , $P(B)=\tfrac{3}{4}$ , and $P(A\cap B)=0.55$ , are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A student is deciding between two study apps that give random scholarship credits each month. The student cares about the expected credits earned per month over many months. Strategy A (App A): - With probability 0.50, earn 8 credits - With probability 0.30, earn 4 credits - With probability 0.20, earn 0 credits Strategy B (App B): - With probability 0.10, earn 30 credits - With probability 0.40, earn 3 credits - With probability 0.50, earn 0 credits Which strategy has the greater expected value in credits per month over many months?
A survey asks a randomly selected student how many pets they have. The possible responses are 0, 1, or 2 pets. Suppose the probabilities are: $P(0)=0.2$ , $P(1)=0.5$ , and $P(2)=0.3$ . Let $X$ be the number of pets the student has. Which table correctly represents the probability distribution of $X$ ?
A fair six-sided die is rolled once. Define the random variable $X$ as the number of dots showing on the top face. Which table correctly represents the probability distribution of $X$ ?
A quality-control team tracks two events for a randomly selected product: event $A$ = "the product passes the initial inspection" and event $B$ = "the product passes the final inspection." Given that $P(A)=0.8$ , $P(B)=\tfrac{3}{4}$ , and $P(A\cap B)=0.6$ , are $A$ and $B$ independent? Justify using probabilities.
A farmer tracks the area of a circular oil spill on a pond as it spreads. The radius increases by about 0.5 meters each minute, and the measured area is recorded. Which type of function is most reasonable for modeling the relationship between time and area? Time (min): 0, 1, 2, 3, 4, 5 Area (m $^2$ ): 3.1, 7.1, 12.6, 19.6, 28.3, 38.5
A student plays a carnival game 100 times and records outcomes: win $0 occurred 52 times, win $2 occurred 33 times, and win $5 occurred 15 times. Let $X$ be the dollar amount won on a play. Using relative frequencies from the observed data as estimated probabilities (which may vary with a different sample), what is the expected value of $X$ based on the empirical distribution?
Daily high temperatures (in $^\circ$ F) for 14 days were: 62, 63, 63, 64, 65, 66, 66, 67, 68, 69, 70, 71, 71, 72. Which statement correctly describes the distribution shown by a histogram using 2-degree bins (62–63, 64–65, 66–67, 68–69, 70–71, 72–73)?
A teacher wants to test whether extra practice problems improve math test scores. The teacher randomly assigns students to one of two groups for two weeks: Group 1 completes 0 extra problems per night, and Group 2 completes 10 extra problems per night. Group 2 has higher average test scores. Can we conclude that doing extra practice problems causes higher test scores? Why or why not?
A bag contains 6 black socks and 4 white socks. Two socks are drawn without replacement. Let event $A$ be "the first sock is white," and event $B$ be "the second sock is white." Given $P(A)=\frac{4}{10}$ and $P(B\mid A)=\frac{3}{9}$ , what is the probability that both events occur, $P(A\cap B)$ ?
A wildlife researcher studies foxes. Event $A$ is "a fox is tagged," and event $B$ is "a fox is observed at night." Based on the information given that $P(A)=0.30$ , $P(B)=0.50$ , and $P(A\cap B)=0.12$ , are $A$ and $B$ independent? Use conditional probability to justify by comparing $P(A\mid B)$ and $P(A)$ .
A company hiring process has two stages. First, a candidate passes the resume screen (event $A$ ). Given that the candidate passes the resume screen, the candidate then passes the interview (event $B$ ). If $P(A)=0.25$ and $P(B\mid A)=0.40$ , what is the probability that both $A$ and $B$ occur, $P(A\cap B)$ ?
A school counselor reviewed records from 300 students and found that students who reported more hours of sleep per night tended to have higher scores on a morning alertness test (a positive association). The counselor did not assign sleep schedules; students chose their own sleep habits. Which statement is the most reasonable conclusion?
A teacher recorded the number of minutes it took students to finish a short quiz: 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 25 Which statement best describes the shape and center of the distribution?
A student applies for a scholarship. First, there is a screening step; $P(A)=0.40$ that the student passes the screening. Then, given that the student passes screening, $P(B\mid A)=0.30$ that they are selected for an interview. What is the probability that the student both passes screening and is selected for an interview, $P(A\cap B)$ ?
A dispatcher recorded the delivery times (in minutes) for 12 orders during a lunch rush: 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 14, 35 How does the outlier affect the mean and median delivery time?

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