Practice Test 33 - Statistics and Probability

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

A bike-sharing program is deciding how to place maintenance crews for flat-tire reports. Based on past data: - On weekdays, 60% of flat-tire reports come from downtown and 40% from suburbs. - On weekends, 35% come from downtown and 65% from suburbs. Forecast: there is a 70% chance tomorrow is a weekday-like demand day and a 30% chance it is a weekend-like demand day (due to a major event). The program can choose one of two crew placements for the whole day: - Place the crew downtown. - Place the crew in the suburbs. Goal: maximize the probability the crew starts the day in the area where most reports will occur. Which strategy is most reasonable given the probabilities?
A library tested two reminder methods for returning books on time. They randomly assigned 50 patrons to Treatment A (text reminder) and 50 patrons to Treatment B (email reminder). The outcome was whether the book was returned on time. Treatment A had 41 on-time returns ( $0.82$ ) and Treatment B had 33 on-time returns ( $0.66$ ). The observed difference in proportions was $0.82-0.66=0.16$ (A − B). A randomization test was performed by shuffling the on-time/late outcomes among all 100 patrons 4000 times under "no treatment effect." In the 4000 shuffles, 12 produced a difference in proportions of at least $+0.16$ (A − B). Which conclusion is most reasonable about the treatment effect (A − B)?
A delivery company fits a line to relate package weight to shipping cost: $y = 1.25x + 8,$ where $x$ is the weight in pounds and $y$ is the predicted shipping cost in dollars. What does the slope represent in this context? (Include units.)
A call center tracked the length of 13 customer calls (in minutes): 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 25. Which conclusion about the data is best supported?
A company randomly assigned 70 employees to two formats for a short training module: Treatment A (interactive) and Treatment B (video). Thirty-five employees were randomly assigned to each format. The mean score on a 20-question follow-up quiz was 16.8 for Treatment A and 15.0 for Treatment B, so the observed difference in means was $16.8-15.0=1.8$ points (A − B). To check whether this could happen by chance under "no effect," the company shuffled the training labels 5000 times and recalculated (A − B). In the 5000 shuffles, 22 simulated differences were $1.8$ . Do the results provide evidence that Treatment A is more effective than Treatment B?
A librarian wants to know whether students prefer e-books or printed books. She uses a random number generator to choose 200 student ID numbers from the school database and emails those students a one-question poll asking for their preference. Which type of study is described, and what does the randomization allow?
In a quality-control report, event $A$ is "a randomly selected package is underweight" and event $B$ is "a randomly selected package has a torn seal." Given that $P(A)=0.10$ , $P(B)=\tfrac{1}{2}$ , and $P(A\cap B)=0.04$ , are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A streaming service models a customer's monthly bill with $y = 5x + 9.$ Here, $x$ is the number of movie rentals in a month (rentals) and $y$ is the monthly bill (dollars). What does the y-intercept represent in this context? Include units in your interpretation.
School context: Two events are defined. Event A: a student submits the science project by the deadline. Event B: a student attends the optional project workshop. The teacher reports: "Overall, 60% of students submitted by the deadline. Among students who attended the workshop, 60% submitted by the deadline." Which statement best describes whether A and B are independent?
A random sample of 100 students reported whether they usually do homework at home or at school, and whether they use paper notes or digital notes. Using the two-way table, are the events A = "student uses digital notes" and B = "student does homework at school" independent? Use the table to justify by comparing $P(A\mid B)$ to $P(A)$ . Two-way table (counts): - At school & Digital: 18 - At school & Paper: 12 - At home & Digital: 42 - At home & Paper: 28 - Row totals: At school = 30, At home = 70 - Column totals: Digital = 60, Paper = 40 - Grand total: 100
Based on the sample space shown, $S=\{(H,H),(H,T),(T,H),(T,T)\}$ for flipping two coins (first flip, second flip), let event $A$ be "exactly one head" and event $B$ be "the first flip is a head." Which expression represents $A\cup B$ (A or B)?
Two classes recorded the number of minutes it took students to finish the same puzzle. Data Set A (Class A times, in minutes): 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18 Data Set B (Class B times, in minutes): 10, 11, 12, 12, 13, 14, 14, 15, 16, 18, 22, 35 Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
A warehouse uses a barcode scanner to verify outgoing packages. The scanner correctly reads a correct label 98% of the time. If a label is incorrect (wrong destination), the scanner flags it as incorrect 92% of the time. Historically, 0.5% of packages leave the packing station with an incorrect label (base rate 0.5%). When the scanner flags a package as "incorrect," a supervisor can choose one of these actions. The goal is to minimize the risk of shipping a mislabeled package, even if that means extra manual checks. Which strategy is most reasonable given the probabilities? - Action A: Ship it anyway, because most packages are labeled correctly overall. - Action B: Do a manual verification whenever the scanner flags "incorrect." - Action C: Only manually verify if two consecutive scans flag "incorrect" (two independent scans). - Action D: Ignore the scanner and manually verify a random 1% sample of all packages.
A bag contains 9 equally likely tokens, each showing a letter and a shape: (A, circle), (A, circle), (A, square), (B, circle), (B, square), (B, square), (C, circle), (C, square), (C, square) Let event $A$ be "the token shows a square," and let event $B$ be "the token shows the letter C." Using the outcomes listed, what is $P(A\mid B)$ ? Give your answer as a fraction.
A fair coin is flipped 2 times. Define the random variable $X$ as the number of tails obtained. Which description correctly defines the random variable $X$ ?
Four books— $W$ , $X$ , $Y$ , and $Z$ —are finalists for a class prize, and the class agrees to pick a winner using a fair random method. Fair means each book has probability $\tfrac14$ of being selected. Which procedure is fair?
Quality control context: Event A is "a shipment arrives on time," and event B is "the shipment used Express shipping." Overall, 90% of shipments arrive on time. Among Express shipments, 98% arrive on time. Which statement best describes whether A and B are independent?
A cereal company wants to estimate the mean fill weight of all cereal boxes produced on a particular day. From that day's production, the quality team randomly selects 50 boxes and measures their fill weights. Which statement correctly describes the population and the sample?
A certain measurement error is known to be approximately normal (symmetric, bell-shaped) with mean $\mu=0$ and standard deviation $\sigma=1.5$ . Approximately what percent of errors are between $-1.5$ and $1.5$ ? (Use the empirical rule.)
A school models the number of books checked out from the library with $y = 15 + 4x,$ where $x$ is the number of weeks since the start of the semester (weeks) and $y$ is the number of books checked out that week (books). What does the y-intercept represent in this context? Include units in your interpretation.

Practice tests