1. Distribution of the Sample Proportion
Section
8.2
Distribution of the Sample Proportion

2. Point Estimate of a Population Proportion
Suppose that a random sample of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The sample proportion, denoted (read “p-hat”) is given by
where x is the number of individuals in the sample with the specified characteristic. The sample proportion is a statistic that estimates the population proportion, p.

3. Sampling Distribution of
For a simple random sample of size n with population proportion p:
The shape of the sampling distribution of is approximately normal provided np(1-p)≥10.
The mean of the sampling distribution of is .
The standard deviation of the sampling distribution of is

4. Sampling Distribution of
When sampling from finite populations, this assumption is verified by checking that the sample size n is no more than 5% of the population size N (n ≤ 0.05N).

5. Parallel Example 4: Compute Probabilities of a Sample Proportion
According to the Centers for Disease Control and Prevention, 18.8% of school-aged children, aged years, were overweight in 2004.
In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight?
Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude?

6. Solution
n=90 is less than 5% of the population size
np(1-p)=90(.188)(1-.188)≈13.7≥10
is approximately normal with mean=0.188 and standard deviation =
In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight?
, P(Z>0.05)= =0.4801

7. Solution
is approximately normal with mean=0.188 and standard deviation =
Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude? ,
P(Z>1.91)= =0.028.
We would only expect to see about 3 samples in 100
resulting in a sample proportion of or more.
This is an unusual sample if the true population
proportion is