Download Unit 8, Module 19: The Sampling Distribution for the Population

Unit 8, Module 19: The Sampling Distribution for the Population Proportion
LEARNING GOALS:
1) The mean of a sampling distribution of sample proportions is equal to the population
proportion,  p  p .
2) The standard deviation of a sampling distribution of sample proportions is known as the
p(1  p)
.
n
3) So long as normal criteria are satisfied ( np  10 and n(1  p)  10 ), the distribution of
standard error and can be found by this formula:  p 
sample proportions is Normal and we can use the properties of the Normal Distribution to
determine how “unusual” a particular sample proportion is.
INTRODUCTION:
We discussed a Gallup poll about global warming in our last lesson. Remember that Gallup
asked adults in over 100 countries if they believe that global warming is a serious threat.
The Gallup poll reported 73% of Latin Americans (people who live in Central or South
America) believe that global warming is a serious threat to themselves and their family.
For this lesson, we will assume this value (73%) represents the true population proportion,
and we will randomly sample 250 Latin Americans. We will ask the people in this sample
if they believe that global warming is a serious threat.
TRY THESE:
1) We ask a sample of 250 people if they believe that global warming is a serious threat.
What is the mean,  p , and the standard error,  p of the sampling distribution of sample
proportions?
2) The sampling distribution represents the probability distribution of all possible sample
proportions, drawn from this population with samples of size 250. Is this sampling
distribution approximately normal? Why or why not? Be sure to explain your answer.
3) Based on the mean and standard error, label the values of a-g on the normal distribution
provided below. The values are one standard deviation apart.
4) Use the empirical rule to find the interval centered at p that contains approximately 95%
of all sample proportions.
5) What sample proportions would you consider unusual? Why? Be sure to explain why.
6) In a random sample of 250 Latin Americans, 175 of the people say that global warming is
a serious threat.
a) Find the sample proportion, p̂ .
b) Find the z-score for this sample proportion.
c) Is this sample proportion unusual? Why or why not? Be sure to write an
explanation.
d) In a random sample of 250 Larin Americans, what is the probability that 175 or
more of the people will say that global warming is a serious threat? You may use
technology to find the answer.
7) In another random sample of 250 Latin Americans, 167 of the people say that global
warming is a serious threat.
a) Find the sample proportion, p̂ .
b) Find the z-score for this sample proportion.
c) Is this sample proportion unusual? Why or why not? Please explain.
d) In a random sample of 250 Latin Americans, what is the probability that 167 or
few of the people will say that global warming is a serious threat?
8) In a random sample of 250 Latin Americans, what is the probability that the number of
people who say that global warming is a serious threat will range from 170 to 175 people?
Be sure to show how you got the probability (or explain, if you use an online calculator).