1. Chapter 8 Lesson 8.3
Sampling Variability and Sampling Distributions
8.3: The Sampling Distribution of a Sample Proportion

2. Let’s explore what happens with in distributions of sample proportions (p). Have students perform the following experiment.
Toss a penny 20 times and record the number of heads.
Calculate the proportion of heads and mark it on the dot plot on the board.
What shape do you think the dot plot will have?
The dotplot is a partial graph of the sampling distribution of all sample proportions of sample size 20.
What would happen to the dotplot if we flipped the penny 50 times and recorded the proportion of heads?

3. Sampling Distribution of p
The distribution that would be formed by considering the value of a sample proportion for every possible different sample of a given size from a population.
We will use:
p for the population proportion
and
p for the sample proportion

4. General Properties for Sampling Distributions of p
Rule 1:
Rule 2:

5. The development of viral hepatitis after a. blood
The development of viral hepatitis after a blood transfusion can cause serious complications for a patient. The article “Lack of Awareness Results in Poor Autologous Blood Transfusions” (Health Care Management, May 15, 2003) reported that hepatitis occurs in 7% of patients who receive blood transfusions during heart surgery. We will simulate sampling from a population of blood recipients.
We will generate 500 samples of each of the following sample sizes: n = 10, n = 25, n = 50, n = 100 and compute the proportion of people who contract hepatitis for each sample.
The following histograms display the distributions of the sample proportions for the 500 samples of each sample size.

6. Are these histograms centered around the true proportion
What happens to the shape of these histograms as the sample size increases?

7. General Properties Continued . . .
Rule 3: When n is large and p is not too near 0 or 1, the sampling distribution of p is approximately normal.
The farther the value of p is from 0.5, the larger n must be for the sampling distribution of p to be approximately normal.
Success/Failure Condition:
If np > 10 and nq > 10, then a normal distribution provides a reasonable approximation to the sampling distribution of p.

8. Blood Transfusions Revisited . . .
Let p = proportion of patients who contract hepatitis after a blood transfusion
p = .07
Suppose a new blood screening procedure is believed to reduce the incident rate of hepatitis. Blood screened using this procedure is given to n = 200 blood recipients. Only 6 of the 200 patients contract hepatitis. Does this result indicate that the true proportion of patients who contract hepatitis when the new screening is used is less than 7%?
To answer this question, we must consider the sampling distribution of p.

9. Is the sampling distribution approximately normal?
Blood Transfusions Revisited . . .
Let p = .07
p = 6/200 = .03
Is the sampling distribution approximately normal?
np = 200(.07) = 14 > 10
nq = 200(.93) = 186 > 10
What is the mean and standard deviation of the sampling distribution?
Yes, we can use a normal approximation.

10. Blood Transfusions Revisited . . .
Let p = .07
p = 6/200 = .03
Does this result indicate that the true proportion of patients who contract hepatitis when the new screening is used is less than 7%?
This small probability tells us that it is unlikely that a sample proportion of .03 or smaller would be observed if the screening procedure was ineffective.
This new screening procedure appears to yield a smaller incidence rate for hepatitis.
P(p < .03) =
.0132

11. Practice
Handout

12. Homework
Pg.519: #8.23,24,25,28,31