Practice Test 32 - 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 random sample of 120 students was surveyed about whether they have a part-time job and whether they participate in a school sport. Using the two-way table, are the events A = "has a part-time job" and B = "plays a school sport" independent? Use the table by comparing $P(A\mid B)$ to $P(A)$. Two-way table (counts): - Rows: Part-time job (Yes, No) - Columns: Plays a sport (Yes, No) Job Yes: Sport Yes 18, Sport No 12 (Row total 30) Job No: Sport Yes 54, Sport No 36 (Row total 90) Column totals: Sport Yes 72, Sport No 48, Grand total 120
2. The quiz scores (out of 20) for a small class were: 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 2 Which statement best describes the shape and center of this distribution?
3. A spinner is spun once. Let event $A$ be "the spinner lands on blue" and event $B$ be "the spinner lands on an even-numbered section." Given that $P(A)=0.40$, $P(B)=\tfrac{1}{2}$, and $P(A\cap B)=0.18$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
4. A district email claims: "Students at our school read more than students at another school." It reports that a random sample of 50 students from School A (our school) reported reading an average of 5.2 hours per week. For School B, the email uses results from an optional online survey posted on School B's website; 50 students responded with an average of 4.1 hours per week. The email concludes School A students read more. Which critique best evaluates the claim?
5. A PE teacher recorded the number of push-ups completed by 11 students in one minute: 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 21. Which plot best represents the data as a dot plot on a number line?
6. A box plot summarizes the number of minutes students spent on a homework assignment. The box plot shows: minimum = 8, $Q_1$ = 10, median = 12, $Q_3$ = 14, maximum = 15, with one additional point plotted at 35 minutes beyond the whisker. Which statement is best supported?
7. A jar contains 5 red marbles and 3 blue marbles. You draw one marble first without replacement (event $A$: the first marble is red). Then you draw a second marble next without replacement (event $B$: the second marble is blue). If $P(A)=\frac{5}{8}$ and $P(B\mid A)=\frac{3}{7}$, what is the probability that both $A$ and $B$ occur (i.e., $P(A\cap B)$)?
8. A basketball team is down by 2 points with 10 seconds left and must decide between two plays. Play 1 (Quick 2): probability of making the 2-point shot is $0.55$. If they miss, probability of getting an offensive rebound is $0.25$; if they get the rebound, probability of scoring on the follow-up is $0.50$. Play 2 (Go for 3): probability of making the 3-point shot is $0.38$. If they miss, probability of getting an offensive rebound is $0.18$; if they get the rebound, probability of scoring a 2-point follow-up is $0.55$. The team's goal is to maximize the probability of taking the lead or tying before time expires. Which strategy is most reasonable given the probabilities?
9. Two cards are drawn from a standard deck with replacement. Let event $A$ be "the first card is a heart" and event $B$ be "the second card is a king." Given that $P(A)=\tfrac{1}{4}$, $P(B)=\tfrac{1}{13}$, and $P(A\cap B)=\tfrac{1}{52}$, are $A$ and $B$ independent? Justify using probabilities.
10. A basketball play has two sequential outcomes. First, the player makes the first free throw (event $A$) with probability $P(A)=0.85$. Then, given the first free throw was made, the probability the player makes the second free throw (event $B$) is $P(B\mid A)=0.90$. What is the probability that both events occur, $P(A\cap B)$?
11. A student can choose between two grading schemes for a course project. Payoff is final project score in points. Strategy A: You take a single presentation. Outcomes: with probability 0.75 you score 80 points; with probability 0.25 you score 40 points. Strategy B: You submit a written report. Outcomes: with probability 0.50 you score 90 points; with probability 0.50 you score 60 points. Over many repetitions, which option would result in a higher average outcome (expected project score)?
12. In a quality-control process, a randomly selected part is first found to be out of specification (event $A$). If it is out of specification, it is then classified as a "major" defect (event $B$). If $P(A)=0.08$ and $P(B\mid A)=0.50$, what is the probability that both $A$ and $B$ occur (i.e., $P(A\cap B)$)?
13. The diameters of ball bearings from a well-calibrated process are approximately normal. Diameters are modeled as $N(\mu=10.00\text{ mm},\,\sigma=0.05\text{ mm})$. Approximately what percent of bearings have diameters between 9.95 mm and 10.05 mm? (Use the empirical rule.)
14. A bag contains many marbles, and one marble is selected at random. Event $A$ is "the marble is blue," and event $B$ is "the marble is large." Given that $P(A)=0.50$, $P(B)=0.30$, and $P(A\cap B)=0.20$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
15. A factory records two events for a randomly selected item: event $A$ = "the item is labeled correctly" and event $B$ = "the item is sealed correctly." Given that $P(A)=0.9$, $P(B)=0.7$, and $P(A\cap B)=0.60$, are $A$ and $B$ independent? Justify using probabilities.
16. A coach tracked sprint times (in seconds) for two groups running the same 100 m sprint. Data Set A: 14.8, 15.0, 15.1, 15.2, 15.3, 15.3, 15.4, 15.5, 15.6, 15.7 Data Set B: 14.1, 14.6, 14.9, 15.0, 15.1, 15.2, 15.3, 15.4, 15.9, 16.4 Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
17. A school social media post says: "Joining a club leads to higher GPA." The post summarizes a survey of 120 students: club members had an average GPA of 3.4, and non-members had an average GPA of 3.0. The post concludes: "Clubs raise GPA by 0.4." Which critique best evaluates the claim?
18. A school counselor reviewed records from 300 students and found that students who reported more hours of sleep per night (quantitative) tended to have higher average quiz scores (quantitative), showing a positive association. The counselor did not assign sleep amounts; the data came from a voluntary survey and existing grade records (observational study). Which statement is the most reasonable conclusion?
19. A retailer uses a two-step fraud filter. First, an order is flagged for review (event $A$). Given that it is flagged, the review team confirms it is legitimate (event $B$). If $P(A)=0.12$ and $P(B\mid A)=0.70$, what is the probability that both $A$ and $B$ occur, $P(A\cap B)$?
20. A student repeatedly chooses between two quiz strategies. The student wants the higher expected score change per quiz (points). Strategy A: With probability 0.50 you gain 12 points; with probability 0.50 you lose 6 points. Strategy B: With probability 0.90 you gain 4 points; with probability 0.10 you lose 20 points. Which strategy has the greater expected value over many repetitions?

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