Use software to solve these problems. The r-code is provided for all calculations. Be sure to
interpret the output in your own words.
Ch 5 homework problems (5 points each)
5.16 a – c
a. Standard deviation = 0.216
b. In 95% of cases, the average total sleep time for college students is between 6.698 and
7.562 hours.
c. P = 0.143
5.30 a – d
a. No; X has more than two possible outcomes. Additionally, the trials might not be
independent. Outcomes are likely to be affected by how active the person is, how much
free time they have, or access to resources, for example. The sample does consist of a
fixed number of students chosen at random; however, X is not a count of successes
since it represents a continuous variable.
b. Yes; X has two outcomes. The shoes are either defective or non-defective. Each trial is
independent, and the sample consists of a fixed number of shoes chosen at random.
The probability of a pair being defective can be considered constant. Therefore, a
binomial distribution is reasonable.
c. Yes; X has outcomes. Students either eat or do not eat at least five servings of fruits or
vegetables a day. Each trial is independent, and the sample consists of a fixed number of
college students chosen at random. The probability of a student saying that they eat at
least five servings can be considered constant. Therefore, a binomial distribution is
reasonable.
d. No; X has more than two possible outcomes. The trials also might not be independent if
you consider sick days or vacation days in which you will likely skip class. Additionally,
there is not a sample to pull from, and X is not a count of successes.
5.64 a – d
a. Mean = 0.91; standard deviation = 5.57
b. The law of large numbers states that the more observations there are, the closer the
sample mean gets to the population mean. As Sam continues to spin, his observed
results will reflect the actual results more and more. The average payoff that Sam
receives from his bets is $0.91, and the standard deviation is $1.76.
c. The central limit theorem states that the larger the sample size, the better the Normal
approximation of the sampling distribution of the sample mean. The mean will become
equal to the expected value of X, and the standard deviation will become equal to the
standard deviation of X divided by the square root of n. Sam’s average payoff after
betting on 520 spins in a year is $0.91, and the standard deviation is $0.24.
d. The probability that the average payoff is greater than 1 is 0.356, and the true
probability is 0.396. This is a reasonable approximation using the central limit theorem.