9.3 Sample Means (pp.591-604)
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Chapter 9.3 Homework
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1. The mean and standard deviation of a population are parameters.
What symbols are used to represent these parameters?
𝜇 𝑎𝑛𝑑 𝜎
2. The mean and standard deviation of a sample are statistics.
What symbols are used to represent these statistics?
𝑥̅ 𝑎𝑛𝑑 𝑠𝑥
3. Because averages are less variable than individual outcomes, what is true about the standard deviation of the
sampling distribution of 𝑥̅ ?
It is smaller than the population standard deviation.
4. What symbols are used to represent the mean and standard deviation of the sampling distribution of 𝑥̅ ?
𝑥̅ 𝑎𝑛𝑑 𝑠𝑥
5. What is the mean of the sampling distribution of 𝑥̅ , if 𝑥̅ is the mean of an SRS of size n drawn from a large
population with mean μ and standard deviation σ?
It is the same as the population mean.
6. What is the standard deviation of the sampling distribution of 𝑥̅ , if 𝑥̅ is the mean of an SRS of size n drawn from a
large population with mean μ and standard deviation σ?
𝜎
√𝑛
7. To cut the standard deviation of 𝑥̅ in half, you must take a sample _4____ times as large.
8. When should you use
𝜎
√𝑛
to calculate the standard deviation of 𝑥̅ ?
When the population is 10 times as large as the sample
9. What does the Central Limit Theorem say about the shape of the sampling distribution of 𝑥̅ ?, no matter what
shape the population distribution has?
As the sample size increases, the sampling distribution will approximate the Normal curve better and better.
10. Investors remember 1987 as the year stocks lost 20% of their value in a single day. For 1987 as a whole, the mean
return of all common stocks on the New York Stock Exchange was 𝜇 = −3.5%. (That is, these stocks lost an average of
3.5% of their value in 1987.) The standard deviation of the returns was about 𝜎 = 26%. Page 592 in your book shows
the distribution of the mean returns 𝑥̅ for all possible samples of 5 stocks.
(a) What are the mean and the standard deviation of the distribution on page 592?
(b) Assuming that the population distribution of returns on individual common stocks is Normal, what is the probability
that a randomly chosen stock showed a return of at least 5% in 1987? Show your work.
(c) Assuming that the population distribution of returns on individual common stocks is Normal, what is the probability
that a randomly chosen portfolio of 5 stocks showed a return of at least 5% in 1987? Show your work.
(d) What percent of 5-stock portfolios lost money in 1987? Show your work.
11. The scores of individual students on the American College Testing (ACT) composite college entrance examination
have a Normal distribution with mean 18.6 and standard deviation 5.9.
(a) What is the probability that a single student randomly chosen from all those taking the ACT scores 21 or higher.
(b) Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the average
(sample mean) score for the 50 students? Do your results depend on the fact that individual scores have a Normal
distribution?
12. The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and
standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal, because a count takes only
whole-number values. An inspector studies 200 square yards of the material, records the number of flaws found in each
square yard, and calculates 𝑥̅ , the mean number of flaws per square yard inspected. Use the central limit theorem
to find the approximate probability that the mean number of flaws exceeds 2 per square yard. Show your work.
13. In response to the increasing weight of airline passengers, the Federal Aviation Administration (FAA) in 2003
told airlines to assume that passengers average 190 pounds in the summer, including clothes and carry-on
baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation
is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and
women, but they are not very non-Normal. A commuter plane carries 20 passengers.
(a) Can you calculate the probability that a randomly selected passenger weighs more than 200 pounds? If so, do
it. If not, explain why not.
(b) Can you calculate the probability that the total weight of the passengers on the flight exceeds 4000 pounds? (Hint:
Restate the problem in terms of the mean weight.) If so, do it. If not, explain why not.