Stats HW 5

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University of Pittsburgh *

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STAT 1000

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Statistics

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Feb 20, 2024

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docx

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Uploaded by ConstableScorpionMaster6886

5.16 Sleep duration of college students. In Example 5.4, the daily sleep duration among college students was approximately Normally distributed with mean μ=7.13 hours and standard deviation σ=1.67 hours. You plan to take an SRS of size n=60 and compute the average total sleep time. What is the standard deviation for the average time? Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean. What is the probability that your average will be below 6.9 hours? 5.30 Should you use the binomial distribution? In each of the following situations, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case. In a random sample of students in a fitness study, X is the mean daily exercise time of the sample. A manufacturer of running shoes picks a random sample of 20 shoes from the production of shoes each day for a detailed inspection. X is the number of pairs of shoes with a defect. A nutrition study chooses an SRS of college students. They are asked whether or not they usually eat at least five servings of fruits or vegetables per day. X is the number who say that they do. X is the number of days during the school year when you skip a class. 5.64 A roulette payoff. A $1 bet on a single number on a casino’s roulette wheel pays $35 if the ball ends up in the number slot you choose. Here is the distribution of the payoff X: Payoff X $0

$35 Probability 0.974 0.026 Each spin of the roulette wheel is independent of other spins. What are the mean and standard deviation of X? Sam comes to the casino weekly and bets on 10 spins of the roulette wheel. What does the law of large numbers say about the average payoff Sam receives from his bets each visit? What does the central limit theorem say about the distribution of Sam’s average payoff after betting on 520 spins in a year? Sam comes out ahead for the year if his average payoff is greater than $1 (the amount he bet on each spin). What is the probability that Sam ends the year ahead? The true probability is 0.396. Does using the central limit theorem provide a reasonable approximation?

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Stats HW 5

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