Test for a Difference in Proportions Section 6.9 Test for a Difference in Proportions

Outline Pooled proportion Test for a difference in proportions

Split or Steal? http://www.youtube.com/watch?v=p3Uos2fzIJ0  http://www.youtube.com/watch?v=p3Uos2fzIJ0 Both choose “split”: split the jackpot Both choose “steal”: both get nothing One “steal,” one “split: stealer gets everything What would you do??? (a) Split (b) Steal Watch the video… pause before the results, and ask students what they would do Van den Assem, M., Van Dolder, D., and Thaler, R., “Split or Steal? Cooperative Behavior When the Stakes Are Large,” 2/19/11.

Split or Steal? MALE FEMALE Total SPLIT 140 163 303 STEAL 129 142 271 269 305 574 Are males or females more cooperative? pM = proportion of males who split pF = proportion of females who split

Split or Steal? Are males or females more cooperative? pM = proportion of males who split pF = proportion of females who split The relevant hypotheses are: H0: pM = pF , Ha: pM > pF H0: pM = pF , Ha: pM < pF H0: pM = pF , Ha: pM ≠ pF H0: pM ≠ pF , Ha: pM = pF

Split or Steal? MALE FEMALE Total SPLIT 140 163 303 STEAL 129 142 271 269 305 574 pM = proportion of males who split pF = proportion of females who split Calculate the sample statistic, 𝑝 𝑀 − 𝑝 𝐹 140/269 – 163/305 = 0.520 – 0.534 = -0.014

Hypothesis Testing SE = 𝑝 1 (1− 𝑝 1 ) 𝑛 1 + 𝑝 2 (1− 𝑝 2 ) 𝑛 2 For hypothesis testing, we want the distribution of the sample proportion assuming the null hypothesis is true H0: p1 = p2 SE = 𝑝 1 (1− 𝑝 1 ) 𝑛 1 + 𝑝 2 (1− 𝑝 2 ) 𝑛 2 What to use for p1 and p2?

Pooled Proportion We assume the proportions are the same between the two groups, and want to use one proportion that is our best guess for what they would be, if they were equal Combine both groups into one big group, and use the overall proportion, called the pooled proportion, 𝒑 Hint: the pooled proportion will always be somewhere in between 𝑝 1 and 𝑝 2

Split or Steal? MALE FEMALE Total SPLIT 140 163 303 STEAL 129 142 271 269 305 574 Pooled proportion = overall proportion who split 𝑝 = 140+163 269+305 = 303 574 =0.528

Test for a Difference in Proportions 𝑧= 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 −𝑛𝑢𝑙𝑙 𝑆𝐸 𝑧= 𝑝 1 − 𝑝 2 𝑝 (1− 𝑝 ) 𝑛 1 + 𝑝 (1− 𝑝 ) 𝑛 2 If 𝑛 𝑝 ≥10 and 𝑛(1− 𝑝 )≥10 for both sample sizes, then the p-value can be computed as the area in the tail(s) of a standard normal beyond z.

Split or Steal? MALE FEMALE Total SPLIT 140 163 303 STEAL 129 142 271 269 305 574 Based on these data, can we conclude whether males or females are significantly more cooperative when playing Golden Balls? Yes No

Split or Steal? Counts are greater than 10 in each category H0: pM = pF , Ha: pM ≠ pF 𝑝 = 140+163 269+305 = 303 574 =0.528 𝑧 = 𝑝 1 − 𝑝 2 𝑝 (1− 𝑝 ) 𝑛 1 + 𝑝 1− 𝑝 𝑛 2 = 0.520−0.534 0.528(1−0.528) 269 + 0.528(1−0.528) 305 = −0.014 0.042 =−0.33 p-value = 2(0.371) = 0.742 Based on these data, we cannot conclude whether males or females are more cooperative.

Split or Steal Actually – it’s much more interesting. Younger males are much less cooperative than younger females, but older males are much more cooperative than older females! (for more practice, students can test for significance within under 30 or over 50)

Difference in Proportions Come up with your own categorical variable that you would like to analyze by gender, using this class. This should be a categorical variable with possible answers: yes or no

Difference in Proportions MALES ONLY ANSWER: Your question here. (a) Yes (b) No

Difference in Proportions FEMALES ONLY ANSWER: Your question here. (a) Yes (b) No

Difference in Proportions Is there a significant difference in the proportion answering “yes” between males and females? (a) Yes (b) No Have them work through the results on their own, and then go over it on the board.