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Midterm

STATS Midterm 2 study.docx

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Department
Statistics
Course
STAT 202
Professor
Dina Dawoud
Semester
Fall

Description
Question 1: Score 0/0 The probability of getting a parking ticket when not paying for a 2-hour period is 0.25. What is the probability of getting at least 75 tickets if you park on 257 occasions for a 2-hour period and don’t pay? Incorrect Your Answer: Correct 0.0699 Answer: Comment: Let X be the number of tickets received. This is a binomial situation wp = 0.25 and n = 257 , soMean = np = 64.25 , . Using a Normal approximation with Continuity Correction we have Thus, P(X ≥ 75) ≈ 1 - P(Z < 1.476579) = 0.0699. Question 2: Score 0/0 Assume that the tires sold by Olsen Tires are normally distributed with a mean life of 41,000 miles and a standard deviation of 2,450 miles. If you were to buy 4 Olsen tires, what is the approximate probability that all four will last longer than 40,000 miles? Incorrect Your Answer: Correct Answer: 0.188 Commen t: Let X = tire lifetime. E(X) = 41,000 anX σ = 2,450. We standardize X by setting . Now consider a single tire, and let's find the probability that its lifetime is 40,000 miles or more. That is 4 The probability that all four tires will last this long, assuming their lifespans are independent, is (0.6584) = 0.188 . Question 3: Score 0/0 If X is a normal random variable with mean 5 and standard deviation 2.9, then find the value x such that P(Z > x) is equal to 0.7263, as shown below. (Note: the diagram is not necessarily to Incorrect scale.) Your Answer: Correct Answer: 3.2551 Comment: P(X > x) = 0.7263 means 1 - P(X < x) = 0.7263 so P(X < x) = 1 - 0.7263 = 0.2737 Standardizing: so using the Inverse Normal we have : Question 4: Score 0/0 If a baseball player's batting average is 0.314 or 31.4%, find the probability that the player will have a bad season and only score at most 67 hits in 235 times at bat. (4 decimal accuracy). NOTE: Please answer with a probability, not a %. For example 0.1234 instead of 12.34 . Incorrect Your Answer: Correct 0.1883±0.001 Answer: Comment: Let X be the number of hits in 235 at-bats. We use the normal approximation to the binomial here. Mean Variance so Question 5: Score 0/0 Suppose at the University of Manitoba, 34.3% of the students live in apartments. If 178 students are randomly selected, then the probability that the number of them living in apartments will be between 49 and 68 inclusive, is (4 decimals): Incorrect Your Answer: Correct 0.8564±0.001 Answer: Comment Let X be the number of students in apartments. : This is a binomial distribution p = 0.343 and n = 178 . Mean = np = 61.054, Var = np(1 - p) = 40.1125 , soSD = 6.3334 Using the normal approximation we continuity correction we have: Question 6: Score 0/0 Defects occur in a certain manufactured tape on the average of 1 per 1,000 m. Assuming a Poisson distribution for the number of defects in a given length of tape, what is the probability that a 4,000 m roll will have no defects? (3 decimal accuracy) Incorrect Your Answer: Correct Answer: 0.0183±0.01 Comment: Let X = number of defects in a 4,000 m roll. X ~ Poisson(4). P(X = 0) = = 0.0183 . Question 7: Score 0/0 A series of n independent trials are run for a Binomial Process with probability of success p. If the mean is found to be 2.9 and the variance is 1.4, what would you estimate n to be? Incorrect Your Answer: Correct Answer: 6 Comment: We have μ = 2.9 and σ = 1.4 . Using the properties of the Binomial Distribution we have : [1] np = 2.9 and [2] np(1-p) = 1.4 Combining : 2.9(1-p) = 1.4 so or = 0.517241 Substitute this p value in [1] and solve:: For n you really should round UP to the next integer, but "normal" roundoff is accepted. Question 8: Score 0/0 The World Series terminates when one team wins its fourth game. Suppose the two teams are evenly matched, so each has probability 1/2 of winning any one game. What is the probability that the series will take 5 games? Incorrect Your Answer: Correct Answer: Comment: We have a binomial situation. Define success as team A winning a game. Then p = = 1 - p and n = 5 . The correct calculation (for the answer ) is shown in red below: P(Series terminates at the end of 4 games) = P(A wins all 4 or B wins all 4) Consider the case where B wins all 4. Then x = 0 and: P(series ends in 4 with B the winner) = P(X = 0; n = 4, p = 0.5) = By symmetry, P(series ends in 4 with A the winner) = , so P(series ends in 4) = P(Series terminates at the end of 5 games) = 2P(A wins in 5) by the symmetry of A and B = 2 P(A wins exactly 3 of the first 4, then fifth) = 2 P(X = 3; n = 4, p = )P(A wins 5th) = 2 P(Series ends after 6th game)= P(A wins in 6 OR B wins in 6)= 2P(A wins in 6) by the symmetry of A and B = 2P(A wins 3 of the first 5)P(A wins 6th) = 2 P(X = 3; n = 5, p = ) = 2 P(Series runs 7 games)= P(even after 6 games) = P(X = 3; n = 6, p = ) = . Question 9: Score 0/0
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