Practice Test 36 - Statistics and Probability

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

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1. A tutoring center wants to know whether online tutoring or in-person tutoring leads to higher algebra test scores. They recruit 80 students who sign up for tutoring and then randomly assign each student to receive either online tutoring or in-person tutoring for four weeks. At the end, all students take the same algebra test. What does the randomization (if any) allow the tutoring center to conclude?
2. A drawer contains 5 black socks and 3 white socks. Two socks are drawn without replacement. Order does not matter because only the colors drawn matter. There are ${8\choose 2}$ equally likely 2-sock selections. What is the probability that the two socks are the same color?
3. A jar is said to contain red marbles with probability $P(\text{red})=0.50$ on each draw (with replacement). You draw 12 times and get 10 reds. You run 60 simulated sets of 12 draws under $P(\text{red})=0.50$. In those simulations, 11 out of 60 runs produced 10 or more reds ("as extreme or more extreme" means $\ge 10$ reds). Which conclusion is most reasonable based on the simulations?
4. A streaming service wants to estimate the proportion of all active subscribers in the United States who watched at least one documentary last week. It randomly selects 500 U.S. active subscriber accounts and finds that 210 watched at least one documentary. What population parameter is being estimated?
5. A school counselor reviewed records from 300 students and found that students who reported more hours of sleep per night tended to have higher scores on a morning alertness test (a positive association). The counselor did not assign sleep amounts; students kept their usual routines and just reported their sleep. Which statement is the most reasonable conclusion?
6. A researcher fits a line $y = 0.6x + 5$ to model a plant's height, where $x$ is the number of days since planting (days) and $y$ is the plant's height (centimeters). What does the y-intercept represent in this context? Include units in your interpretation.
7. A track coach recorded the times (in seconds) for 12 students to run a 100-meter sprint: 12.1, 12.4, 12.4, 12.6, 12.8, 12.8, 12.9, 13.0, 13.2, 13.2, 13.5, 13.7. Which plot best represents the data as a dot plot on a number line?
8. A city compared monthly average outdoor temperature (in $^\circ$C) and monthly electricity use (in kWh) over 5 years and found that hotter months tend to have higher electricity use (a positive correlation). The city did not control temperature; it simply recorded the values. Can we conclude that higher temperature causes higher electricity use? Why or why not?
9. A cafeteria tracked whether students chose pizza or salad for lunch and whether they are in Grade 9 or Grade 10. The results are shown in the two-way table. Which statement best describes the association between grade level and lunch choice?

   	Pizza	Salad	Total
   Grade 9	54	26	80
   Grade 10	36	54	90
   Total	90	80	170

10. A spinner game defines event $A$ as "landing on a multiple of 3" and event $B$ as "landing on a green region." Given that $P(A)=0.30$, $P(B)=\tfrac{2}{5}$, and $P(A\cap B)=0.12$, are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$).
11. A survey asks about exercise. Event $A$ is "a respondent runs at least once per week" and event $B$ is "a respondent swims at least once per week." The probabilities are $P(A)=0.36$, $P(B)=0.29$, and $P(A\cap B)=0.11$. What is the probability that $A$ or $B$ occurs (inclusive or)?
12. A coffee shop models the predicted wait time with $y = 0.6x + 2.5,$ where $x$ is the number of customers in line (customers) and $y$ is the predicted wait time (minutes). What does the slope represent in this context (include units)?
13. A company tracks product features. Event $A$ is "a product has wireless charging," event $B$ is "a product is water-resistant." Data show $P(A)=0.30$, $P(B)=0.50$, and $P(A\cap B)=0.18$. What is the probability that $A$ or $B$ occurs (inclusive "or")?
14. A student is choosing between two raffle ticket bundles to buy repeatedly over many school fundraisers. Each bundle costs money up front and then has one of the listed outcomes. Strategy A (cost $4): - Win $0 with probability 0.70 - Win $10 with probability 0.25 - Win $50 with probability 0.05 Strategy B (cost $6): - Win $0 with probability 0.55 - Win $15 with probability 0.40 - Win $30 with probability 0.05 Considering net winnings (prize minus cost), which strategy has the greater expected value?
15. A phone buyer is deciding whether to purchase a protection plan. Consider the buyer's expected total cost over many similar buyers. Strategy A (Buy plan): Pay $30 now. - 0.90 probability of no damage (no additional cost) - 0.10 probability of damage; repair cost would be $200 but the plan covers it (no additional cost) Strategy B (No plan): Pay $0 now. - 0.90 probability of no damage (no cost) - 0.10 probability of damage and paying the $200 repair Which option would result in a lower average total cost for the buyer over many repetitions?
16. A game is played by drawing one card at random from a set of 5 cards labeled $\{1,1,2,3,3\}$. You win an amount equal to the number on the card. Let $X$ be your winnings (in dollars). Which table correctly represents the probability distribution of $X$?
17. According to $S=\{\text{H},\text{T}\}$ for flipping one coin, let event $A$ be "the outcome is heads" and event $B$ be "the outcome is tails." Which expression represents $A\cap B$ (A and B)?
18. A teacher surveyed 160 students about whether they completed the optional practice set and whether they scored 80% or higher on the quiz. Based on the two-way table, what is the conditional relative frequency (as a percent) of Scored 80% or higher = Yes given that the student completed the practice set?
19. A factory claims 5% of its lightbulbs are defective, so $P(\text{defective})=0.05$. A quality inspector checks 60 bulbs and finds 9 defective. You decide to use simulation: repeatedly generate 60 bulbs with a 0.05 chance of being defective and count how often you get 9 or more defectives ("as extreme or more extreme" means $\ge 9$). If your simulation shows that 0 out of 200 simulated runs had 9 or more defectives, would this result cause you to question the model? Why?
20. Adult heights in a large population are roughly bell-shaped, so using a normal model is reasonable. Suppose heights have mean $\mu=170$ cm and standard deviation $\sigma=6$ cm. Approximately what percent of adults are shorter than 164 cm? (Use the empirical rule.)

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