Practice Test 14 - Statistics and Probability

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

Start practice test

1. A survey records the number of text messages a randomly selected person sends in an hour. Let $X$ be the number of messages. The distribution is: $P(X=0)=0.5$, $P(X=2)=0.3$, $P(X=4)=0.2$. What is the expected value of $X$ (in messages)? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)
2. School context: Event A is "a student is late at least once this week," and event B is "a student rides the school bus." The office reports: 12% of bus riders were late at least once, and 12% of non-bus riders were late at least once. Which statement best describes whether A and B are independent?
3. A bag contains 6 red marbles and 4 blue marbles. One marble is drawn without replacement, and then a second marble is drawn. Let $A$ be "the first marble is red," so $P(A)=\frac{6}{10}$. Let $B$ be "the second marble is blue given the first was red," so $P(B\mid A)=\frac{4}{9}$. What is the probability that both the first marble is red and the second marble is blue (i.e., $P(A\cap B)$)?
4. A teacher claims that on a multiple-choice quiz with 4 options per question, a student who guesses randomly has probability $P(\text{correct})=0.25$ on each question. A student guesses on 20 questions and gets 11 correct. A simulation of 100 runs of 20 random guesses shows that 2 runs had 11 or more correct answers ("as extreme or more extreme" means $\ge 11$ correct). Which conclusion is most reasonable?
5. From 10 students, a 4-person committee is chosen at random. Order does not matter because committee roles are not assigned. There are ${10\choose 4}$ equally likely committees. What is the probability that two specific students, Maya and Noah, are both on the committee?
6. One card is drawn from a standard 52-card deck. Let event $A$ be drawing a red card, and event $B$ be drawing a king. Given $P(A)=\frac{26}{52}$, $P(B)=\frac{4}{52}$, and $P(A\cap B)=\frac{2}{52}$, what is $P(A\cup B)$ (inclusive or: red, king, or both)?
7. A cafeteria manager wants to estimate the population parameter $p$, the proportion of all customers who choose a vegetarian option on a given day. A random sample of 150 customers found 39 chose vegetarian, so the sample statistic is $\hat p = 39/150 = 0.26$ (26%). Students ran a simulation of 1,000 repeated random samples of size 150, and about 95% of simulated sample proportions were between 0.19 and 0.33. Which interval is a reasonable estimate range for the population parameter $p$ based on the simulation? (Use proportions, not percents.)
8. A company tracks product features. Event $A$ is "a randomly selected laptop has a touchscreen" and event $B$ is "it has a backlit keyboard." The probabilities are $P(A)=0.52$, $P(B)=0.46$, and $P(A\cap B)=0.30$. What is the probability that $A$ or $B$ occurs (inclusive or)?
9. A researcher surveyed 500 adults and found that people with higher weekly exercise minutes tended to have lower resting heart rates (a negative association). Participants reported their usual exercise; the researcher did not assign exercise routines. Can we conclude that increasing exercise minutes causes a lower resting heart rate? Why or why not?
10. A small business tracked the number of customer emails received per day for 11 days: 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 60. Which statement best describes the shape and center of the distribution?
11. A deck contains 12 cards: 5 are stars and 7 are circles. You draw one card first without replacement (event $A$: the first card is a star). Then you draw a second card next without replacement (event $B$: the second card is a circle). If $P(A)=\frac{5}{12}$ and $P(B\mid A)=\frac{7}{11}$, what is the probability that both $A$ and $B$ occur (i.e., $P(A\cap B)$)?
12. Three teams—Red, Blue, and Green—are tied and need a tiebreaker to decide who gets the first choice of practice time. A fair method means each team has probability $1/3$ of being selected. Which method gives each team an equal chance?
13. A plant grows according to the model $y = 1.8x + 6,$ where $x$ is time in days since the plant was measured and $y$ is the predicted height in centimeters. Which interpretation of the slope is correct? (Include units.)
14. A raffle will award 1st, 2nd, and 3rd prizes to three different people from 9 entrants. Order matters because 1st/2nd/3rd are different prizes. There are $9P3$ equally likely outcomes. What is the probability that Jordan wins any prize (1st, 2nd, or 3rd)?
15. At a school, two events are defined: Event A = "a student turned in the homework," and Event B = "a student attended the after-school help session." A teacher says, "Among students who attended the help session, 90% turned in the homework. Overall, 70% of students turned in the homework." Which comparison best explains conditional probability here?
16. A survey records the number of minutes a student spends reading on a given day. Let $X$ be the number of minutes. The distribution is: $P(X=10)=\tfrac{1}{4}$, $P(X=20)=\tfrac{1}{2}$, $P(X=30)=\tfrac{1}{4}$. What is the expected value of $X$? Give your answer in minutes.
17. A spinner has 4 equal sections labeled 1, 2, 3, and 4. The spinner is spun once. Let $X$ be the number of points won, defined as follows: if the spinner lands on 1 or 2, you win 0 points; if it lands on 3, you win 1 point; if it lands on 4, you win 3 points. Which table correctly represents the probability distribution of $X$?
18. According to $S=\{(H,1),(H,2),(H,3),(T,1),(T,2),(T,3)\}$ for the experiment "flip a coin (H/T) and spin a 3-section spinner (1/2/3)," define $A$ as "the coin shows H" and $B$ as "the spinner shows an odd number." Which set of outcomes represents event $B$?
19. A school blog claims: "The new library hours increased daily library visits by 50%." The blog reports that average daily visits rose from 200 to 300 after the hours changed. However, the "before" number is the average from one week in September, and the "after" number is the average from the week right before final exams in December. Which critique best evaluates the claim?
20. A community center wants to estimate the population parameter $\mu$, the mean amount of time (in minutes) visitors spend at the center on a weekday. A random sample of $n=25$ visitors had a sample statistic $\bar{x}=74$ minutes. Students used simulation by repeatedly taking 3,000 random samples of size 25 from a model centered at 74 minutes. About 95% of the simulated sample means were between 64 and 84 minutes. Which interval is a reasonable estimate range for the population parameter $\mu$?

Practice tests