Practice Test 46 - Statistics and Probability

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

A spinner has 8 equal sections labeled 1 through 8. One spin is made. The outcomes are equally likely. Let event $A$ be "the result is even," and let event $B$ be "the result is greater than 4." What is $P(A\mid B)$ ? Write your answer as a simplified fraction.
A teacher randomly assigns 4 different students (A, B, C, D) to sit in 4 seats in a row. Order matters because different seatings are different outcomes. What is the probability that A and B sit next to each other (in either order)?
A researcher surveys 120 commuters about how many days per week they work from home. The observed counts are: 0 days (30), 1 day (24), 2 days (36), 3 days (18), 4 days (12). Define the random variable $X$ as the number of work-from-home days for a randomly selected commuter. Treat the relative frequencies from this survey as estimated probabilities. What is the expected value of $X$ based on the empirical distribution?
A jar contains many marbles. One marble is selected at random. Let event $A$ be "the marble is red" and event $B$ be "the marble is large." Given that $P(A)=0.30$ , $P(B)=\tfrac{2}{5}$ , and $P(A\cap B)=0.12$ , are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A teacher wants to find out whether using a spaced-practice app improves quiz scores. The teacher randomly assigns students in her class to either use the app for 10 minutes per day for two weeks or to study as they normally do for two weeks. At the end, all students take the same quiz. Which statement correctly describes what the randomization allows in this study?
A biologist models the mass of a plant (y, in grams) from the number of days since planting (x) using $\hat{y}=2x+10$ . Residuals are $\text{residual}=y-\hat{y}$ . Which statement best describes what the residual pattern suggests about the model choice?
Four students—Ava, Ben, Cara, and Diego—need a fair way to choose who presents first. Fair means each student has an equal probability ( $1/4$ ) of being selected, even if the same person could be chosen on different days. Which procedure is fair?
A spinner has 4 equal sections labeled 1, 2, 3, and 4. The spinner is spun once. Define the random variable $X$ as the amount of money won in dollars, where the winnings equal $2$ times the number landed on. Which description correctly defines the random variable $X$ ?
A school randomly selects a student. Event $A$ is "the student is in the drama club," and event $B$ is "the student takes a language course." Given that $P(A)=0.30$ , $P(B)=\tfrac{1}{2}$ , and $P(A\cap B)=0.15$ , are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$ ).
From 7 different books, a student randomly chooses 4 books to take on a trip. Order does not matter because the books are just being chosen, not arranged. What is the probability that 2 specific books (Book A and Book B) are both included in the selection?
A student randomly selects one of the 9 equally likely ordered pairs shown: $\{(R,1),(R,2),(R,3),(G,1),(G,2),(G,3),(B,1),(B,2),(B,3)\}$ Let event $A$ be "the color is $G$ ," and let event $B$ be "the number is 2 or 3." What is $P(A\mid B)$ ? Write your answer as a simplified fraction.
A city wants to estimate the population parameter $\mu$ , the mean number of minutes adults in the city spend commuting to work. A random sample of $n=40$ adults had a sample statistic (sample mean) of $\bar{x}=31.5$ minutes. Students used simulation by repeatedly taking 2,000 random samples of size 40 from a model centered at 31.5 minutes. In the simulation, about 95% of the simulated sample means fell between 28.5 and 34.5 minutes. What margin of error is supported by the simulation for estimating the population parameter $\mu$ ?
A gym wants to estimate the population parameter $\mu$ , the mean number of days per week all members work out. A random sample of 60 members gave a sample mean (sample statistic) of $\bar x = 3.9$ days. Students used simulation with 2,000 repeated random samples of size 60. The results showed that about 95% of simulated sample means were within 0.4 days of 3.9. Which interval is a reasonable estimate range for the population parameter $\mu$ based on the simulation? (Round endpoints to the nearest tenth.)
A charity models the total amount raised after $x$ days of fundraising using $y = 250x + 1000,$ where $x$ is time (days) and $y$ is total money raised (dollars). What does the slope represent in this context? (Include units.)
A manufacturer wants to estimate the mean lifetime (in hours) of all light bulbs produced on a particular day. From that day's production, the quality team randomly selects 50 bulbs and measures their lifetimes. Which statement correctly describes the population and the sample?
A histogram shows the distribution of package delivery times (in days) for 30 packages. Most packages are between 2 and 5 days, but there is a single package that took 14 days. Histogram counts by bin: 2–3 days: 9 packages 3–4 days: 11 packages 4–5 days: 8 packages 5–6 days: 1 package 13–14 days: 1 package Which statement is best supported by the histogram?
A school counselor wants to estimate the average number of hours of sleep students at Central High get on school nights. From the student roster, the counselor uses a random number generator to select 120 students and emails them a short questionnaire asking, "How many hours did you sleep last night?" No changes are made to students' routines. Which type of study is described, and what does the randomization (if any) allow the counselor to do?
A factory makes two types of phone chargers: Standard (80% of daily output) and Fast (20%). A quick screening test flags a charger as "likely defective." The test correctly flags 90% of truly defective chargers (sensitivity 90%) and incorrectly flags 5% of non-defective chargers (false positive rate 5%). Historical defect rates differ by type: 1% of Standard chargers are defective, and 4% of Fast chargers are defective. A shipment is flagged by the screening test, and the label shows it is a Standard charger. The quality manager must choose a policy for what to do next for flagged Standard chargers. The goal is to minimize unnecessary destructive teardown while still catching most truly defective units. Which strategy is most reasonable given the probabilities? Policy options: - Policy 1: Immediately tear down every flagged Standard charger. - Policy 2: Do a slower confirmatory test on flagged Standard chargers; only tear down if the confirmatory test is also positive. (Confirmatory test has sensitivity 95% and false positive rate 1%.) - Policy 3: Ignore the screening result for Standard chargers and only do random teardowns. - Policy 4: Tear down only flagged Fast chargers, not flagged Standard chargers.
A class has 10 students. Each student is equally likely to be selected at random. The students are: - Soccer: Ana, Ben, Cam, Dee - Basketball: Eli, Fay, Gus - Neither: Hal, Ivy, Jay Let event $A$ be "the selected student plays soccer" and event $B$ be "the selected student plays a sport (soccer or basketball)." Using the list above, what is $P(A\mid B)$ ? Give your answer as a simplified fraction.
In a factory, event $A$ is "a randomly selected item is defective," and event $B$ is "the item comes from Machine 1." Suppose $P(B)=0.25$ and $P(A\cap B)=0.02$ . Based on the information given, what is $P(A\mid B)$ ? Give your answer as a fraction in simplest form.

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