Practice Test 24 - Statistics and Probability

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

Start practice test

1. A set of 12 equally likely outcomes is created by choosing one letter from {X, Y, Z} and one number from {1, 2, 3, 4}. The outcomes are: X1, X2, X3, X4, Y1, Y2, Y3, Y4, Z1, Z2, Z3, Z4. Let event $A$ be "the letter is Y" and event $B$ be "the number is 3 or 4." What is $P(A\mid B)$? Give your answer as a simplified fraction.
2. A researcher randomly assigned 200 participants to either use a new budgeting app or continue with their usual method for 3 months. At the end, the app group had higher average monthly savings (in dollars). Which statement is the most reasonable conclusion?
3. A music club wants to estimate the population parameter $\mu$, the mean number of songs students have downloaded in the past month. A random sample of $n=30$ students was taken, and the sample statistic was $\bar x=18$ songs. A simulation repeatedly took 2,000 random samples of size 30 from a large list of students and recorded $\bar x$ each time. In the simulation, about 95% of the simulated sample means were between 14 and 22 songs. Which interval is a reasonable estimate range for the population parameter $\mu$?
4. A nutritionist modeled the relationship between daily sugar intake $x$ (grams) and an energy score $y$ using $\hat{y}=80-0.2x$. Residuals ($y-\hat{y}$) for 8 people are shown. Which statement best describes how well the model fits the data based on the residual pattern?
5. A school blog reports: "Switching to reusable water bottles reduced plastic waste by 50%." The blog compares the number of plastic bottles found in hallway trash during one week in September (200 bottles) to one week in December (100 bottles). The blog does not mention any other changes at school. Which statement best describes whether the conclusion is justified?
6. Eight players need a fair way to decide who serves first in a game: P1, P2, P3, P4, P5, P6, P7, P8. Fair means each player has probability $1/8$ of being selected. Which procedure is fair?
7. A school store report states: "Reusable water bottles are becoming more popular." It notes that in September the store sold 30 reusable bottles, and in October it sold 45 reusable bottles. Based on this, the report concludes: "Student interest in reusable bottles increased by 50%." Which critique best evaluates the claim?
8. Three teams (Red, Blue, Green) are tied and must be assigned to present in the first time slot. The method is fair only if each team has probability $\tfrac13$ of being chosen for the first slot (fairness is about equal chances, not equal outcomes). Which procedure is fair?
9. A website tracks user behavior. Event $A$ is "a user clicks the newsletter link," and event $B$ is "a user is on a mobile device." The data show $P(A\cap B)=0.14$ and $P(B)=0.35$. Based on the information given, what is $P(A\mid B)$? Give your answer as a fraction in simplest form.
10. A club has 8 members. Three distinct officer positions—President, Vice President, and Treasurer—will be filled by selecting 3 different members. Order matters because the positions are different. What is the probability that Alex is selected as one of the three officers?
11. A basketball player practiced free throws for 200 shots. The observed outcomes were: Made (146 shots) and Missed (54 shots). Define a scoring variable $X$ as follows: $X=2$ for a made free throw and $X=0$ for a missed free throw. Using the relative frequencies from these 200 shots as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
12. Three students—Ava, Ben, and Carlos—must be chosen to present first in class. The teacher wants the choice to be fair, meaning each student has an equal probability of being selected (not that the same student is chosen equally often over time). Which procedure is fair?
13. A store manager recorded the number of customers entering the store each hour for 12 hours: 14, 15, 15, 16, 16, 17, 18, 18, 19, 21, 22, 24. Which statement correctly describes the distribution if shown as a histogram or dot plot?
14. A contest randomly assigns 1st place, 2nd place, and 3rd place among 8 finalists. Because the places are different, order matters. What is the probability that a particular finalist (Jordan) wins exactly 2nd place?
15. Five students—Jae, Kira, Luis, Mei, and Noor—must pick one person at random to be the group leader for today. Fair means each student has probability $\tfrac15$ of being chosen. Which procedure is fair?
16. A principal is curious whether students who participate in after-school tutoring tend to have higher math grades. The principal uses school records to compare current math grades of students who voluntarily attend tutoring with those who do not. No students are assigned to tutoring by the principal. Which type of study is described, and what can be concluded about causation?
17. A town wants to estimate the proportion of households that use curbside recycling. The town randomly selects 100 households from the list of all households and finds that 62 report using curbside recycling. Which statement best describes what can and cannot be inferred from this random sample?
18. Game costs $5. Roll two fair dice; win $6 on sum 7, $1 on even sum, else $0. Find expected net gain.
19. Two bookstores tracked the number of books sold per hour during a promotion. Data Set A (books/hour): 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24 Data Set B (books/hour): 12, 15, 17, 18, 19, 20, 20, 21, 22, 23, 24, 40 Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
20. Seven runners compete in a race. Gold, silver, and bronze medals are awarded to the top 3 finishers. Order matters because gold/silver/bronze are different. There are $7P3$ equally likely podium outcomes. What is the probability that Lina wins a medal (finishes in the top 3)?

Practice tests