Practice Test 47 - Statistics and Probability

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

A gym wants to estimate the mean number of minutes its members spend exercising per visit. It randomly selects 40 member visits from all visits in January and records the exercise minutes for each selected visit. Why is random sampling important in this situation?
A district wants to estimate the proportion of families who prefer email over paper for school announcements. From a list of all families in the district, they take a random sample of 400 families and send a questionnaire asking their preference. Which type of study is described?
A gym uses the linear model $y = 12x + 35$ to predict total monthly cost. Here, $x$ is the number of personal training sessions in a month (sessions) and $y$ is the total monthly cost (dollars). What does the slope represent in this context? Include units in your interpretation.
A warehouse is deciding how to screen incoming shipments for damaged items. Historically, 4% of items arrive damaged. Two screening strategies are available: - Strategy 1 (Quick Scan): correctly flags 70% of damaged items, but also incorrectly flags 8% of undamaged items. - Strategy 2 (Thorough Scan): correctly flags 90% of damaged items, but also incorrectly flags 18% of undamaged items. Each flagged item is pulled for manual inspection, and the warehouse wants to minimize the chance that an item released without manual inspection is actually damaged, even if that means inspecting more items overall. Which strategy is most reasonable given the probabilities?
A streaming service surveys users. Event $A$ is "a user watches comedy," and event $B$ is "a user watches documentaries." The survey finds $P(A)=0.46$ , $P(B)=0.38$ , and $P(A\cap B)=0.16$ . What is the probability that a user watches comedy or documentaries (inclusive)?
A phone battery is draining at a nearly constant rate while playing music. The battery percentage is recorded below. Time (hours): 0, 1, 2, 3, 4, 5 Battery (%): 100, 92, 84, 76, 68, 60 Which type of function best models the relationship between time and battery percentage?
A password is formed by arranging 4 different letters chosen from the 7 distinct letters {A, B, C, D, E, F, G}, with no repeats. Order matters because different sequences are different passwords. There are $7P4$ equally likely passwords. What is the probability that the password starts with A?
A supermarket manager compared weekly data over a year and found that weeks with higher advertising spending (quantitative) tended to have higher total sales revenue (quantitative), a positive association. The manager did not randomly assign ad spending levels; spending varied based on season and promotions (observational study). Which statement is the most reasonable conclusion?
A bag is said to contain 40% blue marbles, so $P(\text{blue})=0.40$ on each draw with replacement. You draw 25 times and get 6 blue marbles. You simulate 100 runs of 25 draws under $P(\text{blue})=0.40$ . In the simulations, 22 out of 100 runs produced 6 or fewer blues. Which conclusion is most reasonable?
Two movies—Movie X and Movie Y—are tied in a vote for what to watch. The group wants a fair tie-breaker, meaning each movie has probability $\tfrac{1}{2}$ of being chosen. Which procedure is fair?
A lab culture doubles in size every hour. The number of cells (in thousands) is recorded below. Time (hours): 0, 1, 2, 3, 4, 5 Cells (thousands): 3, 6, 12, 24, 48, 96 Which type of function best models the relationship between time and the number of cells?
A phone store offers two warranty plans for a $600 phone. Assume the only possible repair event within a year is a single screen break. Strategy A (Buy warranty): Pay $90 upfront. Outcomes: with probability 0.15 the screen breaks and you pay $0 for repair; with probability 0.85 the screen does not break and you pay $0 for repair. Strategy B (No warranty): Pay $0 upfront. Outcomes: with probability 0.15 the screen breaks and you pay $300 for repair; with probability 0.85 the screen does not break and you pay $0 for repair. From the consumer's perspective, over many repetitions, which option would result in a lower average total cost (in dollars)?
According to the sample space $S=\{1,2,3,4,5,6,7,8\}$ for selecting one card labeled 1–8 from a bag, event $A$ is the set of even outcomes and event $B$ is the set of outcomes greater than 5. Which expression represents $A\cap B$ (A and B)?
A city planner fit the model $\hat{y}=200+15x$ to predict daily subway riders (y, in thousands) from the number of downtown events (x). The residuals for x = 0 through 6 events were: x: 0, 1, 2, 3, 4, 5, 6 residual: 2, -1, 1, -2, 0, 2, -2 Which statement best describes how well the model fits the data?
A city's public library system wants to estimate the proportion of all registered library cardholders in the city who prefer e-books over printed books. The library randomly selects 120 cardholders from the full list of registered cardholders and finds that 48 of them prefer e-books. What population parameter is being estimated?
A coin is claimed to be fair: $P(\text{heads})=0.5$ . In 20 flips, you observe 18 heads. You decide to evaluate the claim by simulation: repeat 20-flip experiments many times using a random number generator with $P(\text{heads})=0.5$ and count how often you get 18 or more heads. Would this result cause you to question the model? Why? (Here "as extreme or more extreme" means $\ge 18$ heads out of 20.)
A card is drawn from a standard 52-card deck first (event $A$ : the card is a heart). Without replacing it, a second card is drawn then (event $B$ : the second card is a heart). You are told $P(A)=\tfrac{13}{52}$ and, given the first card was a heart, $P(B\mid A)=\tfrac{12}{51}$ . What is the probability that both $A$ and $B$ occur?
Six students—P, Q, R, S, T, and U—need a fair way to decide who cleans up after a lab. Fair means each student has probability $\tfrac16$ of being chosen. Which method gives each student an equal chance?
Two delivery services recorded the number of packages delivered per driver in a day. Summary statistics: - Data Set A (Service A): mean = 52.0, median = 52.0, $s$ = 4.0, IQR = 5 (roughly symmetric) - Data Set B (Service B): mean = 52.5, median = 50.0, $s$ = 10.5, IQR = 6 (right-skewed with high values) Which statement best compares the typical number of packages and consistency using appropriate measures?
School context: Two events are defined. Event A: a student has perfect attendance this quarter. Event B: a student is in the Chess Club. The principal says, "Of those who are in Chess Club, 30% have perfect attendance." Which statement best describes what "the chance of A given that B" means in this situation?

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