Warm Up In May 2006, the Gallup Poll asked 510 randomly sampled adults the question “Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?” Of these, 60% answered fairly. Construct a 95% confidence interval for the proportion of all U.S. adults who believe the death penalty is applied fairly. Check the conditions and be sure to interpret your interval.

Answer on 447

Ch. 22 – Comparing Two Proportions (Day 1 –Confidence Intervals) Part V – From the Data at Hand to the World at Large

Comparing Two Proportions In this chapter, we will extend the inference procedures (confidence intervals and hypothesis tests) we’ve been performing for single proportions to situations comparing two proportions So far, we have been asking questions like, “Do more than 50% of Americans approve of the president?” Now we will ask questions like, “Is the approval rating of the president different for men and women?”

The Difference Between Groups In a recent Gallup poll (Feb 17), 46% of 1792 males surveyed stated that they approve of the job Obama is doing. 55% of 1835 females expressed their approval. Find a 95% confidence interval for the true difference in presidential approval between men and women. Note: It is fine that the sample sizes are not exactly the same. It is best if they are fairly close in size, but this is not required.

Back to the formula sheet… Remember the confidence interval formula: In this case, the statistic is the difference between the two sample proportions The critical value is the same z * we used for one- proportion intervals So for this problem, so far we have:

What about the standard deviation? Since this interval involves two samples, you should look in this section of your formula sheet: Will use later→

Well, not quite… According to our formula sheet, we will use But remember that we will really be estimating this value using the standard error, since we only have the sample proportion, not p

Back to the problem… Now we have everything we need to construct our interval:

Don’t forget to interpret! We are 95% confident that the true difference (men - women) in the proportion who approve of President Obama’s performance is between -12.2% and -5.8%. OR: We are 95% confident that the true proportion of men who approve of President Obama’s performance is between 5.8% and 12.2% lower than the proportion of women who approve.

Conditions The conditions for this procedure are very similar to the conditions we used when we worked with single- sample proportions, with one new addition – Random Samples Remember, this time we have two! – 10% condition Sample size is < 10% of population size – Independent groups The two samples must be independent from each other – Success/failure condition(large enough n) This is the same as before, but we have to check it for both samples

For this problem… ConditionCheck Random samples Assume each n < 10% N 1792 is <10% of all U.S. men & 1835 is <10% of all U.S. women Independent Groups Assume n 1 p 1 ≥ 10 n 1 (1 – p 1 ) ≥ 10 n 2 p 2 ≥ 10 n 2 (1 – p 2 ) ≥ (.46)≥ (.54)≥ (.55)≥ (.45)≥10

One more question… Based on our confidence interval, does the poll indicate a statistically significant difference in approval rating between men and women? Yes, since the interval does not contain 0 (which would be the difference if the approval ratings were the same)

Homework 22-1 p. 519 #9, 11, 13, 14