1. The Practice of Statistics
Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 9:
9.2 Sample Proportions
Copyright © 2008 by W. H. Freeman & Company

2. Essential Questions
How do you describe the sampling distribution of a sample proportion? (Shape, center and spread)
How do you compute the mean and standard deviation for the sampling distribution of ?
What is the rule of thumb for the use of the standard deviation of ?
What are the conditions that are necessary to use the Normal approximation to the sampling distribution of ?
How do you use the Normal approximation to the sampling distribution of to solve a probability problem involving ?

3. Review from Chapter 8
X is a random variable. It is the number of successes.
If X has a distribution of B(n, p), then
If we multiply each X by a constant has the effect of multiplying the means and the standard deviation by the constant.

4. The Sampling Distribution of a Sample Proportion
We want to estimate the proportion of success in the population.
We take a SRS form the population of interest.
The estimator in the sample proportion of successes:
The count X is a random variable that follows the Binomial Distribution.

5. The Mean and Standard Deviation of a Sample Proportion
We start with the following facts:

7. When to use the formula for the standard deviation of

8. When to use the Normal Approximation for
The Normal approximation is most accurate when ρ is close to ½ and is least accurate when ρ is near 0 or 1.
The following rule of thumb will insure that Normal calculations are accurate for most statistical purposes;

9. Example
If the true proportion of defectives produced by a certain manufacturing process is 0.08 and a sample of 400 is chosen, what is the probability that the proportion of defectives in the sample is greater than 0.10?
Since nρ = 400(0.08) = 32 > 10 and
n(1-ρ) = 400(0.92) = 368 > 10,
it’s reasonable to use the normal approximation.

10. Example (continued)

11. Example
Suppose 3% of the people contacted by phone are receptive to a certain sales pitch and buy your product. If your sales staff contacts 2000 people, what is the probability that more than 100 of the people contacted will purchase your product?
Clearly ρ = 0.03 and = 100/2000 = 0.05 so ^

12. Example - continued
If your sales staff contacts 2000 people, what is the probability that less than 50 of the people contacted will purchase your product?
Now ρ = 0.03 and = 50/2000 = so ^

13. Classwork
Textbook p.589 problem 9.25
P.590 problem 9.26.