The Sampling Distribution of $\hat{p}$ is Approximately Normal

Since $\hat{p}=X/n$, the sampling distribution of $\hat{p}$ looks the same as that of X except for different numbers on the horizontal axis. For n=30 and p=.4, the probability histogram of X and $\hat{p}$ is shown in Figure  5.1.

  

Figure: Probability Histogram of X and $\hat{p}$

Therefore, like the binomial, the sampling distribution of $\hat{p}$ may be approximated by a normal curve with the correct mean and SD.    

Example: Toss a fair coin 50 times. What is the chance of getting 60% or more heads?
Solution: The question is equivalent to ``What is the probability that $\hat{p}$ exceeds .60?'' Using the mean and SD given in ( 5.2) with n=50 and p=.50, the sample proportion is expected to be around .50 give or take $\sqrt{(.5)(1-.5)}/\sqrt{50}$ or .50 give or take .07. With mean .50 and SD .07, the area to the right of .60 (under the normal curve with mean .50 and SD .07) is .0766. Thus, the proportion of heads will exceed .60 fewer than 8 percent of the time.

Exercise 3. If TV World sold 1200 television sets last year, the percentage of sets sold with extended warranties is expected to be around 20%, give or take ________. Estimate the likelihood that it sold warranties with more than 25% of those sets.