Practice Test 29 - Statistics and Probability

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

Start practice test

1. In a student survey, event $A$ is that a randomly selected student plays a school sport and event $B$ is that the student is in the school band. The probabilities are $P(A)=0.40$, $P(B)=0.35$, and $P(A\cap B)=0.15$. What is the probability that $A$ or $B$ occurs (inclusive or: sport, band, or both)?
2. Five students—Jada, Kai, Lina, Omar, and Priya—will be chosen to lead a warm-up. The coach wants the selection to be fair, meaning each student has probability $\tfrac{1}{5}$ of being selected. Which method gives each student an equal chance?
3. An airline is deciding how to schedule a spare aircraft to reduce cancellations caused by mechanical issues. Historical data: - On any given day, Route A has a $6\%$ chance of a mechanical issue that would cancel the flight unless a spare is available. - Route B has a $3\%$ chance of such an issue. - If the spare is positioned for a route that has an issue, it prevents cancellation $90\%$ of the time (sometimes the spare can't be swapped fast enough). The airline can position the spare for Route A or Route B each day. The goal is to maximize the probability of preventing at least one cancellation (not to minimize costs). Which strategy is most reasonable given the probabilities?
4. A band director recorded practice times (in minutes) for 18 students: 15, 18, 18, 20, 20, 22, 24, 25, 25, 26, 28, 30, 30, 32, 35, 35, 38, 40. Which plot best represents the data as a histogram using bins of width 10: 10–19, 20–29, 30–39, 40–49?
5. A meteorologist modeled the relationship between humidity $x$ (%) and a heat index measure $y$ using the linear model $\hat{y}=20+0.6x$. The residual plot below shows points scattered around 0 with no obvious pattern. Which statement best describes how well the model fits the data?​
6. The weights of apples in a large shipment are roughly bell-shaped, so assume a normal model. Apple weights have mean $\mu=150$ g and standard deviation $\sigma=20$ g. Approximately what percent of apples weigh more than 170 g? (Use the empirical rule.)
7. A fair six-sided die is rolled once. Define the random variable $X$ as the number of even results obtained in this one roll (so $X=1$ if the die shows an even number and $X=0$ otherwise). Which table correctly represents the probability distribution of $X$?
8. A lab test is performed in two steps. First, a sample is labeled correctly (event $A$). If it is labeled correctly, it is then processed without contamination (event $B$). Suppose $P(A)=0.92$ and $P(B\mid A)=0.98$. What is the probability that both $A$ and $B$ occur, $P(A\cap B)$?
9. In a study of 300 households, 120 have a pet (event $B$). Of those, 45 also have a garden (event $A\cap B$), where event $A$ is "the household has a garden." Based on the information given, what is $P(A\mid B)$? Give your answer as a percent.
10. A survey asks adults whether they support policy X or policy Y. Event $A$ is that a randomly selected respondent supports policy X, and event $B$ is that the respondent supports policy Y. Suppose $P(A)=0.48$, $P(B)=0.37$, and $P(A\cap B)=0.20$. What is the probability that $A$ or $B$ occurs (inclusive or: supports X, supports Y, or both)?
11. A raffle ticket yields winnings $X$ (in dollars) with probabilities: $P(X=-1)=\tfrac{3}{4}$ and $P(X=5)=\tfrac{1}{4}$. (A negative value means you lose money overall.) What is the expected value of $X$? Give your answer in dollars.
12. A single card is drawn from a standard deck. Let event $A$ be "the card is a heart" and event $B$ be "the card is a spade." Given that $P(A)=\tfrac{1}{4}$, $P(B)=0.25$, and $P(A\cap B)=0$, are $A$ and $B$ independent? Justify using probabilities.
13. A factory uses a quick test to decide whether to rework a batch. About $15\%$ of batches are actually out of spec. - Test X: detects an out-of-spec batch $85\%$ of the time; incorrectly flags an in-spec batch $12\%$ of the time. - Test Y: detects an out-of-spec batch $70\%$ of the time; incorrectly flags an in-spec batch $4\%$ of the time. Reworking a batch is expensive, so the factory wants to minimize unnecessary rework while still ensuring that most out-of-spec batches get reworked (at least $70\%$). Which strategy is most reasonable given the probabilities?
14. A club has 8 students: 5 seniors and 3 juniors. Two students are chosen at random to be President and Vice President. Since the roles are different, order matters. What is the probability that both officers are seniors?
15. A customer support team is deciding how to route incoming emails. Historically, $30\%$ of emails are urgent. Strategy A: Automatically marks an email as urgent if it contains certain keywords. It correctly identifies urgent emails $80\%$ of the time, but incorrectly marks non-urgent emails as urgent $25\%$ of the time. Strategy B: Uses a stricter keyword list. It correctly identifies urgent emails $65\%$ of the time, but incorrectly marks non-urgent emails as urgent only $8\%$ of the time. Marked-urgent emails go to a fast lane with limited capacity. The team's goal is to keep the fast lane from being overloaded (fewer false urgents) while still capturing at least $60\%$ of truly urgent emails. Which strategy is most reasonable given the probabilities?
16. A PE teacher randomly sampled 100 students and recorded grade level (9th or 10th) and whether the student plays a school sport. Using the two-way table, what is $P(\text{plays a sport}\mid\text{10th grade})$? Give your answer as a percent. Two-way table (counts): - 9th grade & Plays sport: 18 - 9th grade & Does not play: 32 - 10th grade & Plays sport: 30 - 10th grade & Does not play: 20 Row totals: 9th 50, 10th 50 Column totals: Plays 48, Does not play 52 Grand total: 100
17. A fair four-sided die numbered 1–4 is rolled twice. The equally likely outcomes are the 16 ordered pairs $(x,y)$ where $x$ is the first roll and $y$ is the second roll. Let event $A$ be "the sum is 6," and let event $B$ be "the first roll is even." What is $P(A\mid B)$? Write your answer as a simplified fraction.
18. A movie theater recorded the number of minutes of previews shown before the feature film for 14 showings: 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 26 How does the outlier (26 minutes) affect the mean and median preview time?
19. A fair coin is flipped twice. Let $X$ be the number of heads obtained. Which table correctly represents the probability distribution of $X$?
20. In a sequential game, a player must first solve a puzzle (event $A$) and then defeat a boss (event $B$). The probability the player solves the puzzle is $P(A)=0.60$. Given the puzzle is solved, the probability the player defeats the boss is $P(B\mid A)=0.50$. What is the probability that both $A$ and $B$ occur?

Practice tests