Practice Test 48 - Statistics and Probability

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

A spinner is advertised as landing on red with probability $P(\text{red})=0.30$ . A student spins it 50 times and gets 28 reds. They simulate 80 runs of 50 spins using $P(\text{red})=0.30$ . In the simulations, 1 out of 80 runs produced 28 or more reds. Which conclusion is most reasonable based on the simulations?
A coach randomly assigned 72 athletes to two warm-up routines before a sprint test: Treatment A (dynamic warm-up) and Treatment B (light jogging). There were 36 athletes per group. The outcome was sprint time in seconds (lower is better). Treatment A mean time = 12.84 s, Treatment B mean time = 12.90 s. The observed difference (A − B) is $-0.06$ s. A randomization test was performed by shuffling the warm-up labels 1,000 times under "no treatment effect." In the simulations, 463 of the 1,000 shuffled differences were at most $-0.06$ (i.e., at least as negative as the observed difference, in the direction favoring A). Which conclusion is most reasonable about whether Treatment A leads to faster times than Treatment B (one-sided, more negative is more extreme)?
A simple game defines a random variable $X$ as the number of points scored in one round. The distribution is: $P(X=1)=0.3$ , $P(X=2)=0.4$ , $P(X=5)=0.3$ . What is the expected value of $X$ (in points)? (Compute $\sum x\,P(x)$ ; the expected value may not be a possible outcome.)
Two classes, Class 1 and Class 2, are tied for which class gets to use the gym first. They want a fair random method. Fair means each class has probability $\tfrac12$ of being chosen. Which procedure is fair?
A teacher compared two sets of test scores (out of 20 points). Class A scores: 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17 Class B scores: 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 20 Which statement best describes the impact of the 20 in Class B compared with Class A?
A gym uses the model $y = 12x + 25,$ where $x$ is the number of months of membership and $y$ is the predicted total amount paid in dollars. What does the y-intercept represent in this context (include units)?
A soccer player practices free kicks. The height of the ball is recorded at different times after it is kicked. The data are shown below. Which type of function best models the relationship between time and height? Time (s): 0, 1, 2, 3, 4, 5 Height (m): 1, 6, 9, 10, 9, 6
A box contains 12 equally likely slips: 5 say "Math," 4 say "Science," and 3 say "History." One slip is chosen. Let event $A$ be "the slip says Science," and let event $B$ be "the slip does not say Math." What is $P(A\mid B)$ ? Write your answer as a simplified fraction.
A city planner models daily water use (y, in thousands of gallons) from the day's high temperature (x, in °F) with $\hat{y}=1.5x-40$ . Residuals are computed as $\text{residual}=y-\hat{y}$ . Which statement best describes how well the model fits the data?
A bag contains 6 equally likely tiles labeled 1, 2, 3, 4, 5, 6. One tile is chosen at random. Let event $A$ be "the number is even" and event $B$ be "the number is greater than 3." What is $P(A\mid B)$ ? Give your answer as a simplified fraction.
A fair six-sided die is rolled and then a fair coin is flipped. The equally likely outcomes are: $\{1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T\}$ Let event $A$ be "the coin shows heads," and let event $B$ be "the die shows a number greater than 4." Which value best represents $P(A\mid B)$ ? Write your answer as a simplified fraction.
A streaming service survey defines event $A$ as "a user watched a comedy this week" and event $B$ as "a user watched a documentary this week." The survey reports $P(A)=0.62$ , $P(B)=0.28$ , and $P(A\cap B)=0.14$ . What is $P(A\cup B)$ , where "or" is inclusive (comedy or documentary or both)?
A streaming service tracks users. Event $A$ is "a user watches a comedy," and event $B$ is "a user has a premium subscription." Based on the information given that $P(A\cap B)=0.12$ and $P(B)=0.20$ , what is $P(A\mid B)$ ? Give your answer as a percent.
A committee of 3 is chosen from 7 people: 4 seniors and 3 juniors. Order does not matter because no roles are assigned. There are ${7\choose 3}$ equally likely committees. What is the probability that the committee has exactly 2 seniors and 1 junior?
A teacher looked at her class records and found that students who spent more minutes per day on a math practice website (quantitative) tended to have higher end-of-unit test scores (quantitative). Students chose how much to practice; the teacher only collected practice logs and test scores (observational study). Which statement best describes what the data show?
A delivery company models delivery time (y, in minutes) from distance traveled (x, in miles) using $\hat{y}=4x+15$ . Residuals are computed as $\text{residual}=y-\hat{y}$ . Which statement best describes how well the model fits the data?
A survey randomly selects one student from a class. The student's grade level is equally likely to be 9th, 10th, 11th, or 12th. Define the random variable $X$ as the grade level number (9, 10, 11, or 12). Which table correctly represents the probability distribution of $X$ ?
A card is drawn from a standard 52-card deck. Event $A$ is "the card is a spade," and event $B$ is "the card is an ace." Given that $P(A)=\tfrac{1}{4}$ , $P(B)=\tfrac{1}{13}$ , and $P(A\cap B)=0.01$ , are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$ .
In a college, event $A$ is that a randomly selected student lives on campus and event $B$ is that the student has a part-time job. Given $P(A)=0.44$ , $P(B)=0.39$ , and $P(A\cap B)=0.16$ , what is the probability that $A$ or $B$ occurs (inclusive or: lives on campus, has a part-time job, or both)?
A random sample of 170 students was asked whether they prefer morning classes or afternoon classes and whether they drink coffee. Using the two-way table, are the events A = "student drinks coffee" and B = "student prefers morning classes" independent? Compare $P(A\mid B)$ to $P(A)$ .

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