2 sample interval proportions sample

Shown with two examples

Always check conditions

Each of the populations must be at least (10) times each of the corresponding sample sizes; and one sample does not influence the other

Random Sample ... randomly selected or randomly assigned

Large Sample Size; Normality

(the sample has at least 10 expected successes and at least 10 expected failures) Independence ... Population at least 10 times

sample size; and each observation has no influence on any other

Pew Survey on Stem Cell Research(found on page 360)

The researchers from the Pew study interviewed two random samples. Both samples, the one taken in 2002 and the one taken in 2009, had 1500 people. In 2002, 645 people expressed support for stem cell research. In 2009, 870 expressed support. These data are summarized in the table below:

2002 2009 Total

Support Stem Cell Research 645 870 1515

Do Not Support 855 630 148

Total 1500 1500 3000

Using Minitab

Question: Ho: p1 = p2Ha: p1 p2Assuming

a= 5%

Our confidence level will be 95 and we want to find out if they are not equal to.

Results from Minitab

Test and CI for Two Proportions

Sample X N Sample p

1 645 1500 0.430000

2 870 1500 0.580000

Difference = p (1) - p (2)

Estimate for difference: -0.15

95% CI for difference: (-0.185377, -0.114623)

Test for difference = 0 (vs not = 0): Z = -8.31 P-Value = 0.000

Analysis: Our main concern is for the

P-value in this example it is = 0.000

Since this is below our 5% confidence interval we are going to reject the null hypothesis

Don’t forget to properly interpret your findings!

Reject Ho. With a p-value of 0.00, and assuming an α = 0.05, we conclude that we do have statistically significant evidence that the proportion stem cell research supporters differs from the national value.

Another example(8.78 found on page 379)

In 2009 a Gallup Poll reported that 40% of people said they have a gun in their home. In the same year, a Pew Poll reported that 33% of people said they have a gun in the home. Assume that each poll used a sample size of 1000.

Do these polls disagree? Test the Hypothesis that the two populations proportions are different. Use a significance level of 0.05.

Using Mini tab we will follow the same steps used in the

previous example

Gallup Poll: 40% out of 1000 is 400 people

Pew Poll: 33% out of 1000 is 330 people

***Note that we want to use based on the word different in the question***

Don’t forget to set your confidence level and alternative tab accordingly

Results: Test and CI for Two Proportions

Sample X N Sample p1 400 1000 0.4000002 330 1000 0.330000

Difference = p (1) - p (2)Estimate for difference: 0.07

95% CI for difference: (0.0279132, 0.112087)Test for difference = 0 (vs not = 0): Z = 3.26 P-Value = 0.001

Interpretation

Reject Ho. With a p-value of 0.001, and assuming an α = 0.05, we conclude that we do have statistically significant evidence that the proportion of gun owners differs from the national value.