Practice Test 16 - Statistics and Probability

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

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1. To study whether background music affects study efficiency, a researcher recruits 60 volunteers from one high school. Each volunteer is randomly assigned to one of three conditions while completing the same 30-minute practice set: no music, instrumental music, or music with lyrics. Time to finish and number correct are recorded. Which type of study is described?
2. A teacher claims a multiple-choice quiz with 4 options per question is being guessed on, so each answer is correct with probability $P(\text{correct})=0.25$. A student answers 40 questions and gets 18 correct. The teacher simulates 60 runs of 40 guessed answers. In the simulations, 0 out of 60 runs produced 18 or more correct answers. Would this result cause you to question the guessing model? Why?
3. A school selects 2 students from a group of 4 girls and 5 boys to represent the school at an event. Order does not matter. What is the probability that both selected students are girls?
4. A student council wants to estimate the proportion of students at their high school who prefer starting school at 9:00 a.m. instead of 8:00 a.m. They use the school roster to select 120 students using a random number generator and email those students a short questionnaire asking their preference. Which type of study is described, and what does the randomization (if any) allow the researchers to do?
5. A streaming service compared number of ads shown per hour and average viewer rating (out of 5) for different video channels. Technology reports $r=-0.52$. Which statement best interprets the value of $r$ for the relationship between ads per hour and average viewer rating?
6. Scores on a standardized test are designed to be approximately normal. Suppose scores are normally distributed with mean $\mu=500$ and standard deviation $\sigma=100$. Approximately what percent of students score below 300? (Use the empirical rule.)
7. A two-step experiment is performed: first choose a letter from {X, Y} and then choose a number from {1, 2, 3}. All 6 outcomes are equally likely. The outcomes are: {X1, X2, X3, Y1, Y2, Y3}. Let event $A$ be "the outcome has number 1" and event $B$ be "the outcome starts with Y." From the outcomes listed, what is $P(A\mid B)$? Give your answer as a simplified fraction.
8. A math department wants to test whether giving students a "worked-example sheet" before homework increases homework accuracy. Two teachers agree to participate. One teacher's class uses the worked-example sheet for a unit, and the other teacher's class does not. The department then compares average homework accuracy between the two classes. No random assignment is used. Which statement is most accurate?
9. A streaming service is testing two recommendation algorithms to reduce user churn. Among new users, about $10\%$ are "high risk" for churn and $90\%$ are "low risk." - Algorithm A reduces churn for high-risk users with probability $0.40$ and for low-risk users with probability $0.05$. - Algorithm B reduces churn for high-risk users with probability $0.30$ and for low-risk users with probability $0.08$. The company's goal is to maximize the probability of reducing churn across all new users (not just within one subgroup). Which conclusion is best supported by the data?
10. A company fits a model for the value of a used laptop: $y = -120x + 900,$ where $x$ is the age of the laptop in years and $y$ is the predicted value in dollars. What does the slope represent in this context? (Include units.)
11. A company tracks product features. Event $A$ is "a sold laptop has a touchscreen," and event $B$ is "a sold laptop has a backlit keyboard." Suppose $P(A)=0.52$, $P(B)=0.46$, and $P(A\cap B)=0.30$. Using inclusive or, what is $P(A\cup B)$?
12. A PE teacher recorded the number of push-ups completed by 12 students in one minute: 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26. Which plot best represents the data as a box plot (using the five-number summary)?
13. A local website's daily visitors increase by about the same percent each day after a marketing campaign. The data are shown below. Day: 1, 2, 3, 4, 5, 6 Visitors: 500, 575, 661, 760, 874, 1005 Which type of function is most reasonable for modeling the relationship between day and visitors?
14. A random sample of 200 students was asked whether they prefer to study alone or in a group and whether they have taken an online course before. Using the two-way table, estimate the probability that a randomly selected student has taken an online course given that the student prefers to study in a group. Give your answer as a percent. Two-way table (counts): - Rows: Study preference (Alone, Group) - Columns: Taken online course (Yes, No) Alone: Yes 70, No 50 (Row total 120) Group: Yes 30, No 50 (Row total 80) Column totals: Yes 100, No 100, Grand total 200
15. A factory uses a two-step quality control process. First, a product is selected at random; $P(A)=0.12$ that it is defective. Then the product is tested; if it is defective, the test flags it as defective with probability $P(B\mid A)=0.90$. What is the probability that both the product is defective and the test flags it as defective (i.e., $P(A\cap B)$)?
16. Two fair coins are flipped. The equally likely outcomes are $\{HH,HT,TH,TT\}$. Define the random variable $X$ as the number of heads obtained. Which table correctly represents the theoretical probability distribution of $X$?
17. A teacher models a student's total score with $y = 8x + 40,$ where $x$ is the number of homework assignments completed (assignments) and $y$ is total course points (points). What does the slope represent in this context? Include units.
18. A cafeteria manager says: "Students prefer the new sandwich." For one lunch period, the manager stood by the register and asked every 5th student in line to answer a question. Of the 60 students asked, 39 chose the new sandwich and 21 chose the old sandwich. The manager concludes: "About 65% of all students at our school prefer the new sandwich." Which statement best describes whether the conclusion is justified?
19. A music store compared the age of a guitar (years) with its selling price (dollars) for several used guitars. Technology reported a correlation coefficient of $r = 0.52$. Which statement best interprets the value of $r$ for the relationship between guitar age and selling price?
20. A carnival game was played 200 times. The outcomes and counts were: win $0 (92 times), win $2 (68 times), win $5 (40 times). Let $X$ be the amount won (in dollars) on a single play. Using the empirical distribution formed by converting counts to relative frequencies (so the probabilities are estimates based on the observed data), what is the expected value of $X$ based on the empirical distribution?

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