Practice Test 27 - Statistics and Probability

19 of the 20 questions below are in this test; the other one can't be scored yet. Answer them in any order, then submit to see your score and every explanation.

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1. A school collected data from 168 students about whether they bring lunch from home or buy lunch at school, and whether they drink water or juice at lunch. The two-way table shows the counts. What is the conditional relative frequency (as a percent) of students who drink water given that the student brings lunch from home?

   	Water	Juice	Total
   Brings from home	54	30	84
   Buys at school	36	48	84
   Total	90	78	168

2. A card is drawn from a standard 52-card deck. Let $A$ be "the card is a heart" and $B$ be "the card is a face card (J, Q, or K)." Given that $P(A)=\tfrac{1}{4}$, $P(B)=\tfrac{3}{13}$, and $P(A\cap B)=0.05$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
3. A library randomly assigned 56 patrons to receive one of two checkout messages at the self-checkout kiosk: Treatment A (message emphasizing due date) and Treatment B (message emphasizing returning items for others). There were 28 patrons per group. The outcome was whether the patron returned the book on time. Results: Treatment A on-time returns = 23/28 (82.1%), Treatment B on-time returns = 16/28 (57.1%). Observed difference in proportions (A − B) is about $0.25$. A randomization test shuffled the 56 return outcomes between groups 5,000 times under "no treatment effect." In the simulations, 15 of the 5,000 shuffled differences were at least as large as $0.25$. Which conclusion is most reasonable about whether Treatment A increases the on-time return rate compared to Treatment B (one-sided, A − B)?
4. A ball is tossed upward from a balcony, and its height is measured at different times. The data are shown below. Which type of function is most reasonable for modeling the relationship between time and height?
5. A game uses a fair spinner with 4 equal sections labeled 1, 2, 3, and 4. The spinner is spun once. Define the random variable $X$ as the winnings in dollars, where the player wins $0, 1, 1,$ or $3$ dollars when the spinner lands on 1, 2, 3, or 4 respectively. Which table correctly represents the probability distribution of $X$?
6. Roll two fair dice. X=1 if sum 7, 2 if sum 5 or 9, 0 otherwise. Find $P(X=2)$.
7. A card is drawn at random from a set of 12 equally likely cards labeled 1 through 12. Let event $A$ be "the number is a multiple of 3" and event $B$ be "the number is even." Using the outcomes in this model, what is $P(A\mid B)$? Give your answer as a simplified fraction.
8. A gym tracked 180 teen members by workout time and whether they prefer cardio or strength training. Based on the two-way table, which value represents the joint relative frequency (as a decimal) of a teen who works out in the Morning and prefers Cardio?
9. A box of candies is said to contain 25% strawberry candies, so $P(\text{strawberry})=0.25$. You randomly select 16 candies (with replacement) and get 7 strawberry candies. You simulate 200 runs of 16 selections under $P(\text{strawberry})=0.25$. In the simulations, 6 out of 200 runs produced 7 or more strawberry candies. What does the simulation suggest about how unusual the observed result is, and whether you should doubt the model?
10. In a video game, two events happen in order. First, you must find a key (event $A$). If you find the key, you then attempt to open a locked door (event $B$). Suppose $P(A)=0.40$ and $P(B\mid A)=0.75$. What is the probability that both $A$ and $B$ occur, $P(A\cap B)$?
11. A cafeteria manager claims: "Most students prefer the new menu." To support this, they surveyed 200 students by standing next to the salad bar during lunch and asking students who walked by to respond. 150 said they prefer the new menu and 50 said they do not. Which critique best evaluates the claim?
12. A company models the cost of producing custom T-shirts with $y = 120 + 7x,$ where $x$ is the number of shirts produced (shirts) and $y$ is the total cost (dollars). What does the y-intercept represent in this context? Include units in your interpretation.
13. A music app compared the number of songs streamed by users in one day. Data Set A is right-skewed with a few heavy users. Data Set B is also right-skewed but less extreme. Summary statistics: - Data Set A: mean $\bar{x}=36.0$, median $=22.0$, standard deviation $s=40.0$, IQR $=18$ - Data Set B: mean $\bar{x}=28.0$, median $=24.0$, standard deviation $s=18.0$, IQR $=12$ Which statement best compares the typical value and variability of Data Set A and Data Set B using appropriate measures?
14. A cafeteria tracked 200 student purchases by drink choice and whether the student bought a dessert. Based on the two-way table, what is the conditional relative frequency (as a percent) of Bought dessert = Yes given that the student chose Water?
15. School context: Event A is "a student is enrolled in AP Biology," and event B is "a student is a senior." The school reports: among seniors, 30% are enrolled in AP Biology; among non-seniors, 30% are enrolled in AP Biology. Which statement best describes whether A and B are independent?
16. A bag contains 5 red marbles and 3 blue marbles. Three marbles are drawn without replacement. Since only the set of marbles drawn matters, order does not matter. What is the probability that exactly 2 of the marbles drawn are red?
17. A teacher tracked 80 students and found that students who spent more minutes per week on a vocabulary app tended to score higher on a vocabulary quiz (positive correlation). Students chose how much to use the app on their own. Which statement best describes what the data show?
18. An ecologist recorded elevation in meters (x) and average annual temperature in °C (y) at several locations in a mountain region. Technology reports $r=-0.88$. Which statement best interprets the value of $r$?
19. A box contains 10 equally likely balls. Each ball is labeled with a color and a number. The outcomes are: R1, R2, R3, R4, R5, B1, B2, B3, B4, B5 Let event $A$ be "the number is odd," and let event $B$ be "the ball is blue." Based on the outcomes listed, what fraction of outcomes in $B$ also belong to $A$? Give your answer as a fraction.
20. A company tracked its employees for one month and found that employees who drank more cups of coffee per day tended to type more words per minute (a positive association). Employees chose how much coffee to drink. Which statement best describes what the data show?

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