Practice Test 17 - Statistics and Probability

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

A car rental company models the total cost of renting a car with $y = 50 + 30x,$ where $x$ is the number of days the car is rented (days) and $y$ is the total cost (dollars). Which interpretation of the slope is correct? Include units in your interpretation.
Four students—Amina, Ben, Carlos, and Dana—must take turns being the first presenter. The teacher wants a fair method, meaning each student has an equal probability ( $1/4$ ) of being chosen (fairness is about equal chances, not about who ends up first). Which procedure is fair?
A city library wants to estimate the proportion of all adult library cardholders in the city who prefer e-books over printed books. The library uses a computer to randomly select 120 adult cardholders from its full list and surveys them; 48 say they prefer e-books. What population parameter is the library trying to estimate?
A cyclist records the total distance traveled during a steady ride. The data are shown below. Time (minutes): 0, 10, 20, 30, 40, 50 Distance (miles): 0, 3, 6, 9, 12, 15 Which type of function best models the relationship between time and distance?
A student council wants to estimate the average number of hours of sleep students at their high school get on school nights. They use a random number generator to select 120 students from the school roster and email each selected student a short questionnaire asking, "How many hours did you sleep last night?" Which type of study is described?
A factory makes metal rods whose lengths are roughly symmetric and bell-shaped, so a normal model is appropriate. The lengths are modeled as $N(\mu=50\text{ cm},\,\sigma=2\text{ cm})$ . Approximately what percent of rods are longer than $52$ cm? (Use the empirical rule.)
A sports scientist models the number of minutes (y) it takes an athlete to complete a training course based on the athlete's age in years (x) using the linear model $\hat{y}=0.4x+12$ . A table of residuals (defined as $\text{residual}=y-\hat{y}$ ) is shown. Which statement best describes how well the model fits the data?
A wildlife volunteer recorded the number of minutes it took to spot a bird at a feeder on 13 mornings: 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 27. Which conclusion about center is best supported?
An ecologist compared the elevation (meters) of 15 hiking locations with the air pressure (kPa) measured at those locations. Technology reported a correlation coefficient of $r = -0.91$ . Which statement best interprets the value of $r$ for the relationship between elevation and air pressure?
Ten names are entered into a raffle. Three distinct prizes (1st prize, 2nd prize, 3rd prize) are awarded to three different people, and order matters because the prizes are different. What is the probability that Harper wins 1st prize and Quinn wins 2nd prize?
At a school, two events are defined: Event A = "a student takes AP Biology," and Event B = "a student is a senior." A counselor reports: "Among seniors, 40 out of 200 take AP Biology." Which statement best describes what "the chance of A given that B" means in this situation?
A bag is labeled as containing red candies with probability $P(\text{red})=0.60$ . You sample 50 candies (with replacement) and observe 28 reds. A student simulates 80 samples of size 50 using $P(\text{red})=0.60$ ; 52 of the 80 simulated samples had 28 or fewer reds. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as getting $\le 28$ reds in 50 draws.)
According to $S=\{1,2,3,4,5,6,7,8\}$ for the experiment "randomly select one number from 1 to 8," let event $A$ be "the selected number is even," and let event $B=\{x\in S\mid x>5\}$ . Which expression represents $A\cap B$ (A and B)?
A student council post says: "Joining a club reduces stress." The post cites a survey of 120 students at one high school: 70 club members reported an average stress rating of 4.8 (out of 10), and 50 non-members reported an average stress rating of 6.1. Which critique best evaluates the claim?
Two fair six-sided dice are rolled (36 equally likely ordered outcomes). Define the random variable $X$ as the sum of the two dice. Which table correctly represents the theoretical probability distribution of $X$ ?
A school store sampled 125 students and recorded whether they bought a snack and whether they paid with cash or card. Using the two-way table, what is $P(\text{paid with card}\mid\text{bought a snack})$ ? Give your answer as a fraction. Two-way table (counts): - Bought snack & Card: 40 - Bought snack & Cash: 35 - Did not buy snack & Card: 20 - Did not buy snack & Cash: 30 Row totals: Bought snack 75, Did not buy 50 Column totals: Card 60, Cash 65 Grand total: 125
A fitness coach randomly assigned 50 clients to Treatment A (interval training) and 50 clients to Treatment B (steady-state training) for 4 weeks. The outcome was the change in resting heart rate (in beats per minute, bpm), where a more negative value indicates a larger improvement. Summaries: Treatment A mean change $=-6.2$ bpm, Treatment B mean change $=-3.1$ bpm. The observed difference in means (A − B) was $-6.2-(-3.1)=-3.1$ bpm. To test for a treatment effect, the coach performed a randomization test by shuffling the treatment labels 1,000 times and recalculating (A − B). In the simulation under "no effect," 6 of the 1,000 shuffled differences were at most $-3.1$ bpm (i.e., as negative as the observed difference). Do the results provide evidence that Treatment A leads to a larger average decrease in resting heart rate than Treatment B?
A box plot summarizes the commute times (in minutes) for a group of employees. One unusually long commute is shown as a separate point. Five-number summary from the box plot: min = 18, $Q_1$ = 24, median = 28, $Q_3$ = 33, max (outlier) = 62 Which statement is best supported by the box plot?
Sports/game context: Event A is "a player makes the free throw," and event B is "the player practiced free throws that morning." A coach says, "Among players who practiced that morning, 8 out of 10 make the free throw." Which statement best describes what "the chance of A given B" means in this situation?
A die-like game piece was rolled 80 times. The observed results were: 1 (10 times), 2 (14 times), 3 (16 times), 4 (12 times), 5 (18 times), 6 (10 times). Let $X$ be the face value rolled. Using the relative frequencies from these data as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?

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