Chapter 7 – SAMPLING DISTRIBUTIONS
Reading Guide - KEY
7.1 WHAT IS A SAMPLING DISTRIBUTION
1. Explain the difference between a parameter and a statistic?
A parameter is a number that describes some characteristic of the population. In statistical practice, the
value of a parameter is usually not known because we cannot examine the entire population.
A statistic is a number that describes some characteristic of a sample. The value of a statistic can be
computed directly from the sample data. We often use a statistic to estimate an unknown parameter.
2. What is sampling variability? the value of a statistic varies in repeated random sampling
3. Explain the difference between μ and x , and between p and ˆp ?
μ ( a parameter) is the mean of the population and x bar (a statistic) is the mean of the
sample; p (a parameter) is the proportion of the population and p hat known as the sample
proportion is used to estimate the unknown parameter p.
4. What is meant by the sampling distribution of a statistic?
The sampling distribution of a statistic is the distribution of values taken by the statistic in all of the
possible samples of the same size from the same population.
5. When is a statistic considered unbiased? A statistic used to estimate a parameter is an unbiased
estimator if the mean of its sampling distribution is equal to the true value of the parameter being
estimated.
6. How is the size of a sample related to the spread of the sampling distribution? The variability of a
statistic is described by the spread of its sampling distribution. This spread is determined primarily by
the size of the random sample. Larger samples give smaller spread. The spread of the sampling
distribution does not depend on the size of the population, as long as the population is at least 10 times
larger than the sample.
7.2 SAMPLE PROPORTIONS (PP. 432-439)
1. In an SRS of size n, what is true about the sampling distribution of ˆp when the sample size
n increases?
As n increases, the sampling distribution of p hat becomes approximately Normal. Before you
perform Normal calculations, check that the Normal condition is satisfied: np ≥ 10 and n(1 − p) ≥ 10.
2. In an SRS of size n, what is the mean of the sampling distribution of ˆp ? The mean of a sampling
distribution is the same thing as the population proportion.
3. In an SRS of size n, what is the standard deviation of the sampling distribution of ˆp ? The standard
deviation of the sampling distribution of p hat is:
as long as the 10% condition is satisfied:
4. What happens to the standard deviation of ˆp as the sample size n increases? It gets smaller.
5. When does the formula
apply to the standard deviation of ˆp ?
As long as the 10% condition is satisfied. The population must be at least 10 times as large as the
sample. i.e.
6. When the sample size n is large, the sampling distribution of ˆp is approximately normal.
What test can you use to determine if the sample is large enough to assume that the
sampling distribution is approximately normal? The Normal Condition test: Whenever n times p and
n(1 – p) are at least 10, then a Normal distribution can be used to approximate the sampling
distribution of p hat.
7.3 SAMPLE MEANS (PP. 442-454)
1. What symbols are used to represent the mean and standard deviation of the sampling distribution of
x bar? μx bar (mean of the sampling distribution of x bar) and x bar (standard deviation of
the sampling distribution of x bar)
2. What is the mean of the sampling distribution of x bar , if x bar is the mean of an SRS of size n
drawn from a large population with mean m and standard deviation s?
The mean of the sampling distribution of x bar is μxbar = μ.
3. What is the standard deviation of the sampling distribution of x bar, if x bar is the mean of an SRS
of size n drawn from a large population with mean m and standard deviation s?
The standard deviation of the sampling distribution of x bar is
.
4. To cut the standard deviation of x bar in half, you must take a sample __4(four)___ times as large.
5. When should you use
10% condition is satisfied:
to calculate the standard deviation of x bar ? As long as the
i.e. only when the population is at least 10 times as large as the sample
6. If the population distribution is Normal, what can be said about the sampling distribution of
x bar? Does sample size matter? If the population distribution is Normal, then so is the sampling
distribution of x bar. This is true no matter what the sample size is.
7. What does the Central Limit Theorem say about the shape of the sampling distribution of
x bar, no matter what shape the population distribution has? As sample size increases, the distribution
of x bar changes shape; it looks less like that of the population and more like a Normal distribution.
When the sample is large enough, the distribution of x bar is very close to Normal. This is true no
matter what shape the population distribution has, as long as the population has a finite standard
deviation .
8. How large does a sample need to be to use the CLT if the distribution of the population is
not normal? At least 30