Practice Test 50 - Statistics and Probability

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

Start practice test

1. A gamer chooses between two loot boxes to open repeatedly. Net payoff is the value of the item received minus the purchase price. Strategy A: Costs $4. Outcomes: with probability $0.70$ you get an item worth $5 (net $1); with probability $0.30$ you get an item worth $0 (net $-4). Strategy B: Costs $4. Outcomes: with probability $0.10$ you get an item worth $30 (net $26); with probability $0.90$ you get an item worth $0 (net $-4). Which strategy would result in a higher average net payoff over many repetitions?
2. A student recorded the heights (in centimeters) of 12 houseplants: 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 20, 21. Which statement correctly describes the distribution shown by a dot plot?
3. A quality-control team inspects items from a production line. Let event $A$ be "a randomly selected item is scratched" and event $B$ be "a randomly selected item has a loose label." Given that $P(A)=0.20$, $P(B)=\tfrac{3}{10}$, and $P(A\cap B)=0.06$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
4. A spinner has outcomes that sometimes land on blue and sometimes land on a number greater than 4. Let $A$ be "the spinner lands on blue" and $B$ be "the spinner lands on a number greater than 4." Given that $P(A)=0.40$, $P(B)=\tfrac{1}{2}$, and $P(A\cap B)=0.18$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
5. A random variable X has mean 7. What is the expected total over 100 independent observations of X?
6. A fitness app company analyzed data from its users and found a negative correlation between daily minutes of exercise (quantitative) and resting heart rate (quantitative): users who exercised more tended to have lower resting heart rates. The company did not assign exercise levels; it only observed self-reported exercise and measured heart rate from wearable devices (observational study). Can we conclude that increasing exercise causes resting heart rate to decrease? Why or why not?
7. A factory inspects a batch of parts in two steps. First, a part is checked for a surface defect (event $A$). If it passes that check, it is then tested for electrical performance (event $B$). Suppose $P(A)=0.80$ and $P(B\mid A)=0.95$. What is the probability that both $A$ and $B$ occur (i.e., the part passes the surface check and then passes the electrical test), $P(A\cap B)$?
8. School context: Two events are defined. Event A: a student signs up for the spring field trip. Event B: a student is in the Environmental Club. The organizer says, "Among Environmental Club members, 45 out of 60 signed up for the field trip." Which statement best describes what "the chance of A given that B" means in this situation?
9. A fair coin is flipped 2 times. Define the random variable $X$ as the number of tails obtained. Which description correctly defines the random variable $X$?​
10. Headline: "New Planner App Boosts Homework Completion by 40%!" A report says 25 volunteers downloaded the app for two weeks. Before the app, they self-reported completing homework on 50% of school nights; after two weeks, they self-reported 70%. There was no comparison group and no random assignment. Which statement best describes whether the conclusion is justified?
11. In a class, event $A$ is "a randomly selected student submitted the homework on time," and event $B$ is "a randomly selected student attended the review session." The instructor reports $P(A)=0.73$, $P(B)=0.41$, and $P(A\cap B)=0.29$. What is $P(A\cup B)$, where "or" is inclusive (on-time homework or review session or both)?
12. An environmental study uses $y = 3.2x + 18$ to predict ozone level. Here, $x$ is temperature (degrees Celsius) and $y$ is ozone level (parts per billion). Which interpretation of the slope is correct? Include units.
13. School context: Two events are defined. Event A: a student is enrolled in AP Biology. Event B: a student plays on a school sports team. A student asks, "Is the chance of being in AP Biology given that someone plays a sport the same as the chance of playing a sport given that someone is in AP Biology?" Which comparison best explains conditional probability here?
14. A fair coin is tossed 3 times. Let $X$ be the number of heads obtained. Which table correctly represents the probability distribution of $X$?
15. Three friends—Noah, Quinn, and Riley—need a fair way to decide who sits in the front seat today. Fair means each person has probability $\tfrac{1}{3}$ of being chosen. Which procedure is fair?
16. A jar contains 3 green candies and 5 yellow candies. Four candies are drawn without replacement, and order does not matter. What is the probability that all 3 green candies are drawn (and the 4th candy is yellow)?
17. A teacher recorded how many times a student raised their hand during each of 40 class periods. The observed counts were: 0 times (9 periods), 1 time (14 periods), 2 times (11 periods), 3 times (6 periods). Let $X$ be the number of hand-raises in a class period. Using the relative frequencies from these 40 periods as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
18. A school survey recorded whether students are in Band and whether they prefer Math or English. Use the two-way table to answer the question. What is the conditional relative frequency of a student preferring Math given that the student is in Band? Give your answer as a percent.

    	Prefer Math	Prefer English	Total
    In Band	36	24	60
    Not in Band	30	60	90
    Total	66	84	150

19. A café manager looked at receipts from the past year and found that on days with higher rainfall (in mm), the café sold fewer iced drinks (number sold), showing a negative correlation. The manager did not control the weather or assign customers to buy anything. Which statement is the most reasonable conclusion?
20. A card is drawn 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)." Given that $P(A)=0.25$, $P(B)=\tfrac{3}{13}$, and $P(A\cap B)=0.05$, are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$).

Practice tests