Practice Test 35 - 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 runner recorded reaction times (in seconds) for 15 starts during practice. Most starts were between 0.18 and 0.26 seconds, but one start was much slower. Reaction times (seconds): 0.18, 0.19, 0.19, 0.20, 0.20, 0.21, 0.21, 0.22, 0.22, 0.23, 0.23, 0.24, 0.25, 0.26, 0.45 Which measure is most affected by the outlier 0.45 seconds?
2. A random sample of 150 students reported whether they prefer team projects or individual projects, and whether they have taken a computer science class (Yes/No). Using the two-way table, are the events A = "student has taken computer science" and B = "student prefers team projects" independent? Use the table to justify by comparing $P(A\mid B)$ to $P(A)$. Two-way table (counts): - Team & CS Yes: 28 - Team & CS No: 42 - Individual & CS Yes: 32 - Individual & CS No: 48 - Row totals: Team = 70, Individual = 80 - Column totals: CS Yes = 60, CS No = 90 - Grand total: 150
3. In a college, event $A$ is "a student lives on campus," and event $B$ is "a student has a meal plan." Based on the information given that $P(A)=0.55$, $P(B)=0.60$, and $P(A\cap B)=0.30$, are $A$ and $B$ independent? Use conditional probability to justify by comparing $P(A\mid B)$ and $P(A)$.
4. A science class measured the pH of 14 water samples: 6.4, 6.5, 6.6, 6.6, 6.7, 6.8, 6.8, 6.9, 7.0, 7.0, 7.1, 7.2, 7.3, 7.4. Which plot best represents the data as a dot plot on a number line marked in tenths from 6.4 to 7.4?
5. A die is rolled once. Let event $A$ = "the result is a multiple of 3" and event $B$ = "the result is even." Given that $P(A)=\tfrac{1}{3}$, $P(B)=\tfrac{1}{2}$, and $P(A\cap B)=\tfrac{1}{6}$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
6. For normal data, mean 68 and SD 4. Estimate the percentile of 75.
7. A company is deciding between two marketing emails. The company earns profit in dollars depending on the response. The company wants the higher expected profit per email sent over many repetitions. Strategy A: With probability 0.05, profit is $200; with probability 0.95, profit is $0. Strategy B: With probability 0.20, profit is $60; with probability 0.80, profit is $5. Which strategy has the greater expected value (average profit) over many trials?
8. A school is deciding whether to use a quick screening quiz to place students into an advanced workshop. Historically, 15% of students would truly benefit from the advanced workshop. The screening quiz has these properties: - Among students who would benefit, it recommends "advanced" 80% of the time. - Among students who would not benefit, it still recommends "advanced" 20% of the time. The school can either: - Use the quiz recommendation directly, or - Require that a student be recommended "advanced" twice by taking two independent versions of the quiz (same accuracy each time) before placing them. Goal: increase the likelihood that a placed student truly benefits, even if fewer students are placed. Which strategy is most reasonable given the probabilities?
9. At a bookstore, event $A$ is "a customer buys fiction" and event $B$ is "a customer buys nonfiction." Store data show $P(A)=0.52$, $P(B)=0.33$, and $P(A\cap B)=0.17$. What is the probability that a customer buys fiction or nonfiction (inclusive or)?
10. A school counselor took a random sample of 150 students and recorded whether each student prefers morning classes or afternoon classes and whether they usually eat breakfast. Using the two-way table, what is $P(\text{Morning} \mid \text{Eats breakfast})$? Give your answer as a fraction in simplest form. Two-way table (counts): - Rows: Class preference (Morning, Afternoon) - Columns: Eats breakfast (Yes, No) Morning: Yes 54, No 21 (Row total 75) Afternoon: Yes 36, No 39 (Row total 75) Column totals: Yes 90, No 60, Grand total 150
11. A runner recorded mile times (in minutes) over 9 training runs: 7.8, 8.0, 8.1, 8.1, 8.2, 8.3, 8.4, 8.5, 11.6 Which measure of spread is most affected by the outlier 11.6 minutes?
12. A bag contains 2 red marbles (R) and 1 blue marble (B). One marble is drawn at random. Define the random variable $X$ as the number of red marbles drawn. Which table correctly represents the probability distribution of $X$?
13. One school has a few very low test scores. Which comparison best describes typical performance despite outliers?
14. A car magazine compared engine size (liters) and highway fuel efficiency (miles per gallon) across several car models. Technology reports a correlation coefficient of $r=-0.73$. Which statement best interprets the value of $r$ for the relationship between engine size and highway fuel efficiency?
15. A customer is considering two extended warranty plans for a laptop over many similar purchases. Payoffs are measured as net dollars to the customer (positive means money saved compared to paying repairs out-of-pocket; negative means money spent). Strategy A (Cheaper plan): Pay $60 now. Outcomes: with probability 0.85 no repair is needed (net payoff $-60$); with probability 0.15 a repair is needed and the plan covers a $400 repair (net payoff $400 - 60 = 340$). Strategy B (Pricier plan): Pay $90 now. Outcomes: with probability 0.85 no repair is needed (net payoff $-90$); with probability 0.15 a repair is needed and the plan covers the $400 repair (net payoff $400 - 90 = 310$). Which strategy has the greater expected value (average net payoff) over many repetitions?
16. School newsletter report: "Students who use the new study-planner app get higher math grades." The article surveyed 40 students who voluntarily downloaded the app and found their average math grade was 86. It compared this to the schoolwide average math grade of 80 from last semester. The newsletter concludes: "Using the app increases math grades by 6 points." Which critique best evaluates the claim?
17. A teacher compared students' heights (cm) with their math test scores (points) in a class. Technology reported a correlation coefficient of $r = 0.06$. Which statement best interprets the value of $r$ for the relationship between height and math test score?
18. The results of an online placement test are approximately normal (roughly symmetric and bell-shaped). Scores are modeled as $N(\mu=100,\,\sigma=15)$. Approximately what percent of test-takers score above $115$? (Use the empirical rule.)
19. An electronics retailer is deciding how to handle fraud screening for online orders. The fraud model flags an order as "high risk." - Among truly fraudulent orders, 80% are flagged. - Among legitimate orders, 6% are flagged. Only 0.8% of all orders are truly fraudulent. For flagged orders, the retailer can choose: 1) Automatically cancel (no fraud loss, but cancels some legitimate orders). 2) Hold for manual review (catches most fraud, but some fraud slips through; assume manual review misses 10% of fraudulent flagged orders and incorrectly rejects 2% of legitimate flagged orders). 3) Ship immediately (no delay, but fraud loss occurs on fraudulent orders). The retailer's goal is to minimize the chance of wrongly canceling legitimate orders while still reducing fraud relative to shipping immediately. Which strategy is most reasonable given the probabilities?
20. In a student survey, event $A$ is "a randomly selected student plays a school sport" and event $B$ is "a randomly selected student is in the school band." The survey results show $P(A)=0.40$, $P(B)=0.35$, and $P(A\cap B)=0.15$. What is $P(A\cup B)$, where or is inclusive (A, B, or both)?

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