Practice Test 23 - Statistics and Probability

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

Start practice test

1. A spinner is spun once. Let $A$ be "the spinner lands on red" and $B$ be "the spinner lands on an odd number." Given that $P(A)=0.35$, $P(B)=\tfrac{2}{5}$, and $P(A\cap B)=0.12$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
2. Based on the sample space shown, $S=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)\}$, a spinner with outcomes $\{1,2\}$ is spun and then a spinner with outcomes $\{1,2,3\}$ is spun; the ordered pair (first, second) is recorded. Event $A$ is the event that the sum is 4. Event $B=\{(2,1),(2,2),(2,3)\}$. Which expression represents $A\cap B$ (A and B)?
3. In a randomized experiment, 80 campers were randomly assigned to one of two hydration reminders during a hike: Treatment A (phone alert every 20 minutes) or Treatment B (alert every 60 minutes). There were 40 campers in each group. The outcome was whether the camper finished the hike without reporting a headache. Results: 30/40 in Treatment A reported no headache (75%), and 22/40 in Treatment B reported no headache (55%). The observed difference in proportions (A − B) is $0.20$. A randomization test was done by shuffling the 80 outcomes among the two groups 2,000 times under "no treatment effect." In the simulations, 410 of the 2,000 shuffled differences were at least as large as $0.20$. Based on the randomization test, is the observed difference surprising under no effect (in the direction A − B)?
4. A hospital recorded the wait time (in minutes) for 10 patients in a clinic: 12, 13, 13, 14, 14, 15, 15, 16, 16, 45. How does the outlier affect the mean, median, and range?
5. In a school, event $A$ is "a randomly selected student is taking Algebra," and event $B$ is "a randomly selected student is taking Biology." Suppose $P(A)=0.48$, $P(B)=0.44$, and $P(A\cap B)=0.22$. Using inclusive or (Algebra or Biology or both), what is $P(A\cup B)$?
6. A random sample of 160 students was surveyed about whether they have taken an art class this year (Yes/No) and whether they prefer reading or watching videos when learning something new. Using the two-way table, what is $P(\text{Art Yes}\mid\text{Reading})$? Give your answer as a fraction. Two-way table (counts): - Reading & Art Yes: 35 - Reading & Art No: 45 - Videos & Art Yes: 30 - Videos & Art No: 50 - Row totals: Reading = 80, Videos = 80 - Column totals: Art Yes = 65, Art No = 95 - Grand total: 160
7. A student flips two fair coins (coin 1 and coin 2). The sample space is $\{HH, HT, TH, TT\}$. Define the random variable $X$ as the number of heads. Which table correctly represents the probability distribution of $X$?
8. A spinner is spun once. Let event $A$ be "the spinner lands on green," and event $B$ be "the spinner lands on a multiple of 3." Given that $P(A)=\tfrac{3}{10}$, $P(B)=0.20$, and $P(A\cap B)=0.06$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
9. A school surveyed 130 students about whether they attended a tutoring session and whether they turned in the homework on time. Based on the two-way table, which statement best describes the association between attending tutoring and turning in homework on time?
10. A teacher says: "Students who sit in the front row participate more." She observed one class period and tallied participation. Front row (6 students): 18 total comments. Other seats (24 students): 24 total comments. She concludes: "Sitting in the front causes more participation." Which critique best evaluates the claim?
11. A fair six-sided die is rolled once (outcomes $1$–$6$ equally likely). Define the random variable $X$ by the explicit mapping: $X=1$ if the outcome is $1,2,$ or $3$; $X=4$ if the outcome is $4$ or $5$; and $X=10$ if the outcome is $6$. Which table correctly represents the theoretical probability distribution of $X$?
12. A website login process has two sequential checks. First, a user enters a correct username (event $A$), which happens with probability $P(A)=0.90$. Then, given the username is correct, the user enters the correct password (event $B$) with probability $P(B\mid A)=0.80$. What is the probability that both events occur, $P(A\cap B)$?
13. A factory produces metal rods whose lengths are roughly symmetric and bell-shaped, so a normal model is appropriate. Rod lengths have mean $\mu=50$ cm and standard deviation $\sigma=2$ cm. Approximately what percent of rods are longer than 54 cm? (Use the empirical rule approximation.)
14. A quality-control process checks items in two stages. First, an item is chosen from Batch 1 (event $A$) with probability $P(A)=0.25$. Then, given it is from Batch 1, it is defective (event $B$) with probability $P(B\mid A)=0.08$. What is the probability that both events occur, $P(A\cap B)$?
15. A card is drawn first from a standard 52-card deck and kept out (no replacement). Event $A$ is "the first card is a heart," and event $B$ is "the second card is a heart." Given $P(A)=\frac{13}{52}$ and $P(B\mid A)=\frac{12}{51}$, what is the probability that both events occur, $P(A\cap B)$?
16. A school wants to see whether sitting in the front of the classroom improves attention during lectures. For one month, students are randomly assigned seats: half are randomly assigned to front-row seats and half to back-row seats. Teachers then rate each student's attention using the same rubric. Which type of study is described, and what does the randomization allow?
17. A bakery fits the model $y = 4x + 30,$ where $x$ is the number of cupcakes ordered (cupcakes) and $y$ is the predicted total price (dollars). The model was fit using orders between 10 and 100 cupcakes. Is the y-intercept meaningful in context? Choose the best statement.
18. School context: Two events are defined. Event A: a student is in the drama club. Event B: a student takes Spanish this semester. The school counselor says, "Taking Spanish doesn't change the chance that a student is in drama club." Which statement best describes whether A and B are independent?
19. A spinner is spun once. Let event $A$ be "the spinner lands on green" and event $B$ be "the spinner lands on a number greater than 4." Given that $P(A)=0.6$, $P(B)=\tfrac{2}{5}$, and $P(A\cap B)=0.24$, are $A$ and $B$ independent? Justify using probabilities.
20. A delivery service claims that a package arrives on time with probability $P(\text{on time})=0.90$. Over 50 deliveries, a customer reports only 38 arrived on time. A simulation of 100 runs of 50 deliveries each (using $P(\text{on time})=0.90$) found that 0 out of 100 runs had 38 or fewer on-time deliveries ("as extreme or more extreme" means $\le 38$ on time). What does the simulation suggest?

Practice tests