Comparing Two Population Proportions with Confidence Interval

The two “Data” circles in the diagram either represent two separate populations (observational study) or treatment versus control (experiment).

Complete the table below for each pair of samples.

Sample Proportion of pierced females (n=10) Proportion of pierced males (n=5) Difference of proportions (female – male)
1 0.9 0.4
2 0.7 0
3 0.6 0.4 0.2
4 0.8 0.2 0.6
5 0.5 0.4 0.1
6 0.7 0 0.7
7 0.3 0.6 -0.3
8 0.4 0.4 0
9 0.5 0.6 -0.1

Complete the dot plot. The dot plot shows the last seven differences from the table. Complete the dot plot by entering the dots for the first two differences in column 4.

There’s a dot at 0.7. What does this dot represent?

If the difference is zero, what does that say about the two sample proportions?

If the difference is negative, what does this say about proportions of women and men who are pierced?

If the difference is positive, what does this say about proportions of women and men who are pierced?

Calculate the mean and standard deviation of these twelve differences. Write down these values and explain what they mean in the context of the problem.

Based on the few samples we’ve collected, what do you think the difference in population proportions will be? In other words, for the populations of male and female students, what do you think is the difference in proportions pierced?

We used an applet to simulate the random assignment of 16 men to one of two treatments. We examined the difference in proportions of males assigned to each treatment.
In an experiment 10 men are assigned to Treatment 1 and 6 men are assigned to Treatment 2. What is the difference in sample proportions of males assigned to the two treatments? (Treatment 1 – Treatment 2)

Here is a dot plot with simulated random assignments for 32 experiments. Plot the difference you calculated in (a) in the dot plot.

Explain what a positive difference, a negative difference, and a difference of 0 mean in this context.

Here is a dot plot with simulated random assignments for 800 experiments. Explain what the mean and standard deviation tell us.
How does this simulation show that random assignment avoids bias (which is the systematic favoring of one outcome over another)?

To investigate the urban myth that women text more frequently than men while driving, we take a sample of 50 women and 65 men. We find that 14 of the women text while driving and 22 of the men text while driving.
Is this an experiment or an observational study? Explain.

What is the proportion of women in this sample who text while driving? What is the proportion of men in this sample who text while driving?

Find the difference of the sample proportions (women minus men) to two decimal places.

What does it mean if the difference is positive?

If a sample of women text less frequently than a sample of men, would you expect the difference of the sample proportions to be positive or negative?

In a recent class of 1,000 students at StatCrunchU, 49% carried student loans. You think that this percentage is consistent between men and women (the percentage of men with student loans is 49% and the percentage of women with student loans is 49%). You take a sample of 20 men and 15 women. Of the men 12 have student loans and of the women 7 have student loans. Is this typical or unusual if the actual percentage for both is 49%.

In our class group on StatCrunch, look at either colleges4cw2prop.csv or Colleges4CW2prop.csv. Is the proportion of public colleges and proportion of private colleges the same in the population? Why or why not?

Comparing Two Population Proportions with Confidence Interval
Find a confidence interval (CI) for the difference of two population proportions, p_1-p_2, by drawing random samples from each population.
Even if two population proportions are equal, the sample proportions drawn from these populations are usually _____________.
Confidence interval are one method for determining whether different sample proportions reflect ____________________ differences in the populations.
Interpretation of an CI of two population proportions:
If 0 is in the interval, it suggests:

If 0 is NOT in the interval, then:

Interpreting a Confidence Interval for the Difference of Two Population Proportions
Signs Interpretation
Contains 0: (-,+)

Both negative: (-,-)

Both positive: (+,+)

Conditions to check:
1.

2.

3.

4.
CI: point estimate ±z^* 〖SE〗_est: p ̂_1-p ̂_2±z^* 〖SE〗_est

Computation with technology:
Example 6: In August 2014 the Pew Poll asked random samples of 2002 Americans (non-scientists) and 3748 scientists if the Space Station has been a good investment for the US. Of the non-scientists polled, 64% said that the Space Station has been a good investment; of the scientists polled, 68% said that the Space Station has been a good investment. Construct a 95% confidence interval for the difference between the proportion of scientists and non-scientists who believe the Space Station has been good investment for the US. Based on your confidence interval, is there a difference in the population proportions? Explain.
Check conditions:

Calculation:
Interpretation:

Example 7: Extracting natural gas through hydraulic fracturing, or “fracking,” is controversial practice. Residents in two states with large natural gas deposits, Michigan and Pennsylvania, were polled by the University of Michigan to determine whether they favored or opposed fracking. Let p1 represent the proportion of Michigan residents who favor fracking and p2 represent the proportion of Pennsylvania residents who favor fracking. A 95% confidence interval for the difference in the population proportions is (–0.0184, 0.117). Does this interval indicate there is a difference in support of fracking between these two states?

Inference procedure, such as a CI, generalizes a __________ (sample statistic/population parameter) to a ____ (sample/population).
Two ways to reduce margin of error (width of the CI) are: _________ sample size and/or ____________ confidence level.