Confidence Intervals and Two Proportions Presentation 9.4

Two Proportions When examining two proportions, we are interested in the difference between the two. For example, –Is the proportion of males who vote different than the proportion of females who vote? –Statistically, we would estimate the difference between the proportions –That is look at p males - p females

From 1 proportion to 2 For one proportion, we had the formula For two proportions, we look at the difference (so we subtract the proportions and use rules of variances for the standard error)

Assumptions The formula is appropriate when: –The samples are independently selected random samples –In order to use the standard error formula, make sure: population>10n for both samples –In order to approximate with the normal distribution, make sure: n 1 p 1  10n 2 p 2  10 n 1 (1- p 1 )  10 n 2 (1- p 2 )  10

Example: Gender and Voting In a survey done in 2000 about voting and gender, it was found that 106 of 200 males voted in a recent election while 124 of 220 females voted. What is the difference between the proportion of females who vote and the proportion of males who vote? That is, construct a 95% confidence interval for the difference.

Example: Gender and Voting First, find the proportions: p females = 124/220 = call this proportion p 1 p males = 106/200 = 0.53 call this proportion p 2 Now check assumptions: –Check to use standard error. Each of the populations are greater than 10n –There are more than 2200 females. –There are more than 2000 males. –Check for normal approximation. n 1 p 1  10n 2 p 2  (.5636)  10200(.53)    10 n 1 (1- p 1 )  10 n 2 (1- p 2 )  (.4364)  10200(.47)   1094  10 Notice that these are essentially counts of yeses or nos to the potential question (Do you vote?)

Example: Gender and Voting Once the formula is justified, just plug in and go to work. Notice the difference, at the 95% level, includes 0.

Example: Gender and Voting Constructing the interval on the calculator. Choose 2-Proportion Z-interval Enter your statistics Calculate to obtain results.

Confidence Intervals The crucial value to look for in a 2-proportion interval of p 1 -p 2 is zero. –If the interval is all positive, it implies that proportion p 1 is greater than p 2. –If the interval is all negative, it implies that proportion p 1 is less than p 2. –If the interval contains zero, it means these is no significant difference between the proportions. This result is rather important to conducting two sided significant tests for two proportions.

Confidence Intervals and Two Proportions This concludes this presentation.