Practice Test 20 - Statistics and Probability

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

A delivery service models the total delivery fee with $y = 3.5x + 8,$ where $x$ is the distance traveled (miles) and $y$ is the total fee (dollars). What does the $y$ -intercept represent in this context? (Include units in your interpretation.)
A fair six-sided die is rolled once. All outcomes $\{1,2,3,4,5,6\}$ are equally likely. Define the random variable $X$ as the number of factors of the result (including 1 and itself). Explicit mapping from outcomes to $X$ : - $1\mapsto 1$ (factors: $\{1\}$ ) - $2\mapsto 2$ (factors: $\{1,2\}$ ) - $3\mapsto 2$ (factors: $\{1,3\}$ ) - $4\mapsto 3$ (factors: $\{1,2,4\}$ ) - $5\mapsto 2$ (factors: $\{1,5\}$ ) - $6\mapsto 4$ (factors: $\{1,2,3,6\}$ ) Which table correctly represents the theoretical probability distribution of $X$ ?
At a school, two events are defined: Event A = "a student is absent on Friday," and Event B = "a student is on the soccer team." The attendance office reports that absences happen about 10% of the time overall, and also about 10% of the time among soccer players. Which statement best describes whether A and B are independent?
A jar contains 5 red marbles and 3 blue marbles. You draw one marble first without replacement (event $A$ : first marble is red). Then you draw a second marble (event $B$ : second marble is blue). Given that the first marble was red, $P(B\mid A)=\tfrac{3}{7}$ . Also, $P(A)=\tfrac{5}{8}$ . What is the probability that both $A$ and $B$ occur?
A teacher randomly assigned 90 students to two different note-taking methods during a lecture: Treatment A (guided notes) and Treatment B (blank paper). There were 45 students in each group. After the lecture, students took a 20-question quiz. Treatment A mean = 16.8, Treatment B mean = 15.1, so the observed difference (A − B) is $1.7$ questions. A randomization test shuffled the 90 quiz scores between groups 1,000 times under "no treatment effect." In the simulations, 3 of the 1,000 shuffled differences were at least as large as $1.7$ . Which conclusion is most reasonable about the treatment effect (one-sided, A − B)?
An investor is comparing two one-day trading strategies. Payoff is net profit in dollars. Strategy A: With probability 0.60 you gain $50; with probability 0.40 you lose $30. Strategy B: With probability 0.30 you gain $120; with probability 0.70 you lose $10. Over many repetitions, which strategy has the greater expected value (higher average net profit)?
A card is chosen at random from the 16 equally likely cards labeled: A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4, D1, D2, D3, D4. Event $B$ : the letter is C or D. Event $A$ : the number is 1. Based on the outcomes listed, which value best represents $P(A\mid B)$ ? Give your answer as a simplified fraction.
A company uses a non-diagnostic email filter to flag potentially malicious emails. Only 1% of incoming emails are truly malicious. Filter X: - Flags 95% of malicious emails. - Incorrectly flags 4% of non-malicious emails. Filter Y: - Flags 85% of malicious emails. - Incorrectly flags 1% of non-malicious emails. Flagged emails are quarantined for review, and the IT team's goal is to maximize the chance that a quarantined email is actually malicious (so reviewers aren't overwhelmed by false alarms). Which conclusion is best supported by the data?
A fair six-sided die is rolled once; outcomes $\{1,2,3,4,5,6\}$ are equally likely. Define the random variable $X$ as follows: $X=0$ if the roll is 1 or 2; $X=5$ if the roll is 3, 4, or 5; and $X=10$ if the roll is 6. (So $\{1,2\}\to 0$ , $\{3,4,5\}\to 5$ , $\{6\}\to 10$ .) What is the expected value of $X$ based on this theoretical model?
School context: Event A is "a student turns in the homework on time," and event B is "a student attended the after-school help session." The teacher says: "Among students who attended the help session, 18 out of 20 turned in the homework on time." Which statement best describes what "the chance of A given B" means in this situation?
A two-step online checkout process works as follows: a customer must first successfully submit payment (event $A$ ), and then the system must successfully generate a receipt (event $B$ ). The probability payment submission succeeds is $P(A)=0.95$ . Given payment succeeded, the probability a receipt is generated is $P(B\mid A)=0.98$ . What is the probability that both $A$ and $B$ occur?
A bag contains 6 red marbles and 4 blue marbles. Three marbles are drawn without replacement, and order does not matter. What is the probability of drawing exactly 2 red marbles and 1 blue marble?
A teacher fits a linear model to predict a student's test score from the number of hours studied: $y = 8x + 52,$ where $x$ is hours studied (hours) and $y$ is the predicted test score (points). Which interpretation of the y-intercept is correct? (Include units.)
A city collected paired data on daily high temperature in °F (x) and daily electricity use in megawatt-hours (y) for 30 summer days. Technology reports $r=0.06$ . Which statement best interprets the value of $r$ ?
From a standard 52-card deck, event $A$ is "the card is a heart," and event $B$ is "the card is a face card (J, Q, or K)." Suppose $P(A)=\frac{13}{52}$ , $P(B)=\frac{12}{52}$ , and $P(A\cap B)=\frac{3}{52}$ . Using inclusive or, what is $P(A\cup B)$ ?
A prize pays $10 with probability 0.25, otherwise $0. What ticket price makes expected net gain zero?
Which set of statistics most likely comes from data with a low outlier?
A raffle ticket has the following net winnings (in dollars). Let $X$ be the net winnings from one ticket: $P(X=-1)=\tfrac{3}{4}$ , $P(X=3)=\tfrac{1}{4}$ . What is the expected value of $X$ in dollars? (It may not be an outcome.)
Two students, Hana and Owen, both want the last available laptop. They agree to use probability to be fair. Fair means each student has probability $\tfrac12$ of getting the laptop. Which procedure is fair?
Counts 10,15,15 for values 0,1,2. Which is the probability histogram?

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