Practice Test 2 - 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 student recorded the number of texts they received each day for 12 days: 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 12, 14. Which statement correctly describes the distribution when shown as a dot plot?
2. A school wants to estimate the proportion of all students in the school who usually bring lunch from home. The school randomly selects 100 students from the full enrollment list and asks each one. Which statement correctly describes the population and the sample?
3. Scores on a certain standardized test are approximately normal (roughly bell-shaped) with mean $\mu=70$ and standard deviation $\sigma=10$. Approximately what percent of scores are above 90? (Use the empirical rule.)
4. A card is drawn from a standard 52-card deck. Let event $A$ = "the card is an ace" and event $B$ = "the card is black." Given that $P(A)=\tfrac{1}{13}$, $P(B)=0.5$, and $P(A\cap B)=\tfrac{1}{26}$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
5. Heights of adult men in a large region are often modeled as approximately normal. Assume heights are normally distributed with mean $\mu=70$ inches and standard deviation $\sigma=3$ inches. Approximately what percent of men are below 67 inches? (Use the empirical rule.)
6. A teacher modeled the relationship between number of absences $x$ and course grade $y$ using $\hat{y}=95-3x$. For a student with $x=6$ absences, the residual was $+4$ (where $\text{residual}=y-\hat{y}$). Which interpretation is correct?
7. Two groups (Group X and Group Y) both want the larger classroom for their meeting, and the teacher will use probability to decide. The decision is fair if each group has probability $\tfrac12$ of getting the room. Which procedure is fair?
8. Using the outcomes listed in $S=\{\text{A},\text{B},\text{C},\text{D},\text{E},\text{F}\}$ for selecting one letter, let event $A=\{x\in S\mid x\text{ is a vowel}\}$ and event $B=\{\text{B},\text{D},\text{F}\}$. Which set of outcomes represents $A\cup B$ (A or B)?
9. From a standard 52-card deck, event $A$ is "the card is red," and event $B$ is "the card is a king." You are given $P(A)=\tfrac{26}{52}$, $P(B)=\tfrac{4}{52}$, and $P(A\cap B)=\tfrac{2}{52}$. What is $P(A\cup B)$, where or is inclusive?
10. A school newspaper wants to know whether students who participate in a music program tend to spend more time studying. Reporters interview 100 students who happen to be in the library after school, asking (1) whether they are in a music program and (2) how many hours they studied last week. No one is assigned to join music or to study more. Which type of study is described?
11. A teacher recorded whether students turned in homework on time and whether they attended a review session. Use the two-way table to answer the question. What is the conditional relative frequency of turning in homework on time given that the student attended the review session? Give your answer as a percent.

    	On Time	Not On Time	Total
    Attended Review	54	18	72
    Did Not Attend Review	40	48	88
    Total	94	66	160

12. In a student club poll, event $A$ is "a student is in the robotics club" and event $B$ is "a student is in the debate team." The poll shows $P(A)=0.25$, $P(B)=0.18$, and $P(A\cap B)=0.05$. What is the probability that a student is in robotics or debate (inclusive or)?
13. School context: Event A is "a student participates in class," and event B is "a student sits in the front half of the room." In one class, 55% of front-half students participate and 35% of back-half students participate. Which statement best describes whether A and B are independent?
14. A coach tracked 132 students by whether they play a school sport and whether they prefer watching sports or playing sports in their free time. The two-way table shows the counts. What is the conditional relative frequency (as a decimal) of students who prefer watching sports given that the student plays a school sport?

    	Prefer watching	Prefer playing	Total
    Plays a school sport	24	56	80
    Does not play	28	24	52
    Total	52	80	132

15. Students in a school were asked whether they usually do homework at home or at the library and whether they have a part-time job. The results are shown in the two-way table. Which value represents the joint relative frequency (as a decimal) of a student who has a part-time job and usually does homework at the library?

    	Homework at Home	Homework at Library	Total
    Part-time Job: Yes	30	45	75
    Part-time Job: No	60	15	75
    Total	90	60	150

16. A bookstore tracked the number of minutes customers spent browsing (one day sample): 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 16, 35 Which statement best describes the shape and center of the distribution?
17. A single fair six-sided die is rolled once. Let $X$ be the number shown. The distribution is $P(X=k)=\tfrac{1}{6}$ for $k=1,2,3,4,5,6$. What is the expected value of $X$? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)
18. Two students are selected one after the other with replacement from a large school. Let $A$ be "the first selected student is in the art club" and $B$ be "the second selected student is in the art club." Given that $P(A)=0.30$, $P(B)=\tfrac{3}{10}$, and $P(A\cap B)=0.09$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
19. A pet supply store wants to estimate the mean amount (in dollars) spent per purchase among all purchases made at the store this weekend. The store's register system randomly selects 40 receipts from all weekend receipts and computes a sample mean of $18.50. Which statement best describes how the sample is used?
20. The quiz scores (out of 20) for 16 students are: 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 16, 17, 18, 19. Which statement correctly describes the distribution shown by a histogram with equal-width bins?

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