Practice Test 30 - Statistics and Probability

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

A student guesses on a multiple-choice question with 4 answer choices (A–D). Exactly one choice is correct, and each choice is equally likely to be selected when guessing. Define the random variable $X$ as the score where $X=3$ points for a correct answer and $X=-1$ point for an incorrect answer. The mapping is: $\{\text{correct}\}\to 3$ and $\{\text{incorrect}\}\to -1$ (3 outcomes). Which table correctly represents the theoretical probability distribution of $X$ ?
A small publisher is deciding how to print a new book when demand is uncertain. Based on past launches: - There is a $30\%$ chance demand is High. - There is a $70\%$ chance demand is Low. Two printing strategies: - Strategy Large: If demand is High, there is a $0.90$ chance they avoid stockouts; if demand is Low, there is a $0.35$ chance they end up with large unsold inventory. - Strategy Small: If demand is High, there is a $0.55$ chance they avoid stockouts; if demand is Low, there is a $0.10$ chance they end up with large unsold inventory. The publisher's goal is to minimize the chance of large unsold inventory while still keeping the chance of avoiding stockouts at at least $60\%$ overall. Which strategy is most reasonable given the probabilities?
A park service wants to estimate the population parameter $\mu$ , the mean number of minutes visitors spend on a particular trail. A random sample of 25 visitors had a sample mean (sample statistic) of $\bar x = 52$ minutes. A simulation of 1,000 repeated random samples of size 25 produced a distribution of simulated sample means whose middle 95% was from 46 to 58 minutes. What margin of error is supported by the simulation for estimating the population parameter $\mu$ ?
A clinic uses a two-step screening process. First, a patient is selected; $P(A)=0.05$ that the patient has a certain condition. Then, given the patient has the condition, the screening test is positive with probability $P(B\mid A)=0.88$ . What is the probability that both the patient has the condition and the test is positive (i.e., $P(A\cap B)$ )?
In a city, event $A$ is "a randomly chosen day is rainy," and event $B$ is "a randomly chosen day is windy." Based on the information given that $P(A)=0.40$ , $P(B)=0.25$ , and $P(A\cap B)=0.10$ , are $A$ and $B$ independent? Use conditional probability to justify by comparing $P(A\mid B)$ and $P(A)$ .
A model for a child's height is $y = 6x + 52,$ where $x$ is age (years) and $y$ is height (inches). Is the y-intercept meaningful in context, and which statement is best?
A coin is flipped and then a number cube (1–4) is rolled. All outcomes are equally likely. Outcomes (coin, number): (H,1) (H,2) (H,3) (H,4) (T,1) (T,2) (T,3) (T,4) Let event $A$ = "the number is odd" and event $B$ = "the coin shows H." From the outcomes listed, what is $P(A\mid B)$ ? Give your answer as a fraction.
A teacher recorded the number of minutes it took students to finish a short quiz. The times (in minutes) are: 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 16, 17, 29 Which statement best describes the shape and typical value (center) of this distribution?
A meteorologist fit the linear model $\hat{y}=12+0.6x$ to predict afternoon temperature (y, in °C) from morning temperature (x, in °C). The residuals show this pattern: For small x values, residuals are mostly positive; for medium x values, residuals are near 0; for large x values, residuals are mostly negative. What does the residual pattern suggest about the model choice?
A town wants to estimate the proportion of all households in the town that have a backyard garden. A researcher randomly selects 150 addresses from the town's address database and finds that 45 of the selected households report having a backyard garden. Which statement is a reasonable inference?
A card is drawn from a standard 52-card deck. Event $A$ is "the card is a heart," and event $B$ is "the card is a face card (J, Q, or K)." Given that $P(A)=\tfrac{1}{4}$ , $P(B)=\tfrac{3}{13}$ , and $P(A\cap B)=\tfrac{3}{52}$ , are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A student council wants to estimate the population parameter $p$ , the proportion of all students at the school who support adding more vegetarian options in the cafeteria. A random sample of $n=80$ students gave a sample statistic $\hat{p}=0.60$ . Students simulated repeated sampling in two ways: - For $n=80$ , about 95% of simulated sample proportions were between 0.49 and 0.71. - For $n=320$ , about 95% of simulated sample proportions were between 0.55 and 0.65. Based on the simulation results, which margin of error is reasonable for the $n=320$ sample when estimating the population parameter $p$ ?
A password is created by arranging 4 different letters chosen from A, B, C, D, E, F (no repeats). Order matters because different arrangements form different passwords. If one password is chosen uniformly at random from all such 4-letter passwords, what is the probability that it starts with A?
Two classmates, Jada and Malik, both want the last open seat by the window. They agree the process is fair if each has probability $\tfrac12$ of getting the seat. Which procedure is fair?
A gym models the monthly cost of a membership with the linear model $y = 12x + 25,$ where $x$ is the number of months (months) and $y$ is the total cost (dollars). What does the slope represent in this context? (Include units in your interpretation.)
At a school, two events are defined: Event A = "a student signs up for the spring field trip," and Event B = "a student turned in the permission slip by the early deadline." A teacher says, "Of those who turned in the permission slip early, 95% signed up for the trip." Which statement best describes what "the chance of A given that B" means in this situation?
A gym wants to estimate the population parameter $\mu$ , the mean number of times per week all members visit the gym. A random sample of $n=40$ members was selected, and the sample statistic was $\bar x=3.8$ visits per week. A simulation repeatedly took 2,000 random samples of size 40 from the full membership database and recorded $\bar x$ each time. In the simulation, about 95% of the simulated sample means were between 3.2 and 4.4. What margin of error is supported by the simulation for estimating $\mu$ ?
A school compared the number of pages students read over a weekend. Data Set A: 12, 14, 15, 16, 16, 17, 18, 18, 19, 20, 21 Data Set B: 5, 8, 10, 12, 14, 15, 16, 18, 22, 30, 45 Data Set B is right-skewed with a very large value (45). Which statement best compares the typical number of pages read and the variability using appropriate measures (median and IQR)?
Two chips numbered 1,2,3 are drawn with replacement. Let X be the larger number. Which distribution is correct?
A track coach recorded the 12 times (in seconds) it took students to run a 100-meter sprint: 12.4, 12.7, 12.7, 12.9, 13.1, 13.1, 13.2, 13.4, 13.6, 13.6, 13.8, 14.0. Which statement correctly describes the distribution shown when these data are displayed in a dot plot on a number line?

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