Section 8.3
The Sampling Distribution of the
Sample Proportion
The Sample Proportion
• The objective of many statistical investigations is to
draw a conclusion about the proportion of individuals
or objects in a population that possess a specified
property.
• One that has the property of interest is labeled a
success (s), and the ones that do not are labeled a
failure (f).
• For example, what proportion of students are wearing
jeans could be answered using the sample proportion.
Properties of the Sample
Proportion
• The letter π denotes the proportion of S’s
in the population. It is a number
between 0 and 1 (times 100 for %).
• The letter p is the sample proportion of
S’s.
• p = the number of S’s in the sample
n
Properties of the Sampling
Distribution of p
• Let p be the proportion of successes in a random
sample of size n from a population whose proportion of
S’s (successes) is π.
• Denote the mean of p by μp and the standard deviation
by σp. Then the following rules hold:
– Rule 1: μp = 
– Rule 2: σp =
 (1   )
n
– Rule 3: When n is large and  is not too near 1 or 0, the
sampling distribution of p is approximately normal.
Conditions for Using Rule #3
• The farther the value of  is from .5, the
larger n must be for a normal approximation
to the sampling distribution of p to be
accurate.
• A conservative rule of thumb is that if both
n• > 10 and n(1- ) >10, then it is safe to use a
normal approximation.
Ex: The article “Unmarried Couples More Likely to Be
Interracial” (San Luis Obispo Tribune, March 13, 2002)
reported that for unmarried couples living together, the
proportion that are racially or ethnically mixed is 7%
• A random sample of n = 100 couples will be
selected from this population and p, the
proportion of unmarried couples that are
mixed racially or ethnically will be
computed.
• Compute the mean and the standard
deviation of the sampling distribution of p.
Ex: The article “Unmarried Couples More Likely to Be
Interracial” (San Luis Obispo Tribune, March 13, 2002)
reported that for unmarried couples living together, the
proportion that are racially or ethnically mixed is 7%.
• A random sample of n = 100 couples will be
selected from this population and p, the
proportion of unmarried couples that are
mixed racially or ethnically will be
computed.
• Is it reasonable to assume that the sampling
distribution of p is approximately normal for
random samples of size n = 100? Explain.
Ex: The article “Unmarried Couples More Likely to Be
Interracial” (San Luis Obispo Tribune, March 13, 2002)
reported that for unmarried couples living together, the
proportion that are racially or ethnically mixed is 7%.
• Suppose that the sample size is n = 200
rather than n = 100.
• Does the change in sample size change the
mean and standard deviation of the
sampling distribution of p?
• If so, what are the new values for the mean
and standard deviation? If not, explain why
not.
Ex: The article “Unmarried Couples More Likely to Be
Interracial” (San Luis Obispo Tribune, March 13, 2002)
reported that for unmarried couples living together, the
proportion that are racially or ethnically mixed is 7%.
• Suppose that the sample size is n = 200
rather than n = 100.
• Is it reasonable to assume that the sampling
distribution of p is approximately normal for
random samples of size n = 200? Explain.
Ex: The article “Unmarried Couples More Likely to Be
Interracial” (San Luis Obispo Tribune, March 13, 2002)
reported that for unmarried couples living together, the
proportion that are racially or ethnically mixed is 7%.
• When n = 200, what is the probability that
the proportion of unmarried couples in the
sample who are racially or ethnically mixed
will be greater than .10?
Assignment
• p. 427; 27ace, 28-30, 32-33
• Quiz next week on Thursday
• Complete any reading missed this week.