1. 12.2 Comparing Two Proportions

2.  Compare two populations by doing inference about the difference between two sample proportions.

3.  we want to compare two populations or the responses to two treatments based on two independent samples.  We notate our data as follows: PopulationPopulation Parameter Sample size Sample Proportion 1p₁n₁ 2p₂n₂

4.  Example 1: To study the long term effects of preschool programs for poor children, the Ed. Research Foundation has followed two groups of Michigan children since early childhood. One group of 62 attended preschool as 3 and 4 year olds. This is a sample from population 2, poor children and attended preschool. A control group of 61 children from the same area and similar backgrounds represents population1, poor children with no preschool. Thus, the sample sizes are n₁= 61 and n₂=62. A response variable is the need for social services as adults. In the past ten years, 38 of the pre-school sample and 49 of the control sample have needed social services. Therefore: =49/61= 38/62

5.  The Sampling distribution of The mean of is p₁-p₂ The variance of the difference is the sum of the variances of and, which is: ***** WE ADD THE variances, NOT THE standard deviation. When the samples are large, the distribution of is approximately normal.

6.  Draw an SRS of size n₁ from a population having proportion p₁ of successes and draw an independent SRS of size n₂ from another population having proportion p₂ of successes. When n₁ and n₂ are large, an approximate level C confidence interval for p₁- p₂ is:

7.  The standard error SE is: SE= z* is the upper (1-C)/2 standard normal critical value.

8.  Step 1: Show assumptions -they’re independent -both are random sample -both population is at least 10x sample -n₁p₁≥10n₁(1-p₁)≥10 49≥1012≥10 n₂p₂≥10n₂(1-p₂)≥10 38≥1024≥10 PopulationPop. Description Sample Size Sample Proportion 1Control61.803 2Preschool62.613

9.  Step 2: Calculate SE-  Step 3: Calculate the 95% Confidence Interval:

10.  Example 2: While her husband spent 2½ hours picking out new speakers, a statistician decided to determine whether the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. The population was Saturday afternoon shoppers. Out of 67 men, 24 said they enjoyed the activity. 8 of the 24 women surveyed claimed to enjoy the activity. Are the results of the survey significant? Also, give a 99% confidence level of the difference in proportions.

11.  p₁=true proportion of men who enjoy shopping for electronic equipment  p₂= true proportion of women who enjoy shopping for electronic equipment  H₀: p₁=p₂  Ha: p₁>p₂  Assumptions: -they’re independent -both random sample -both population is at least 10x sample 24≥5 8≥5 43≥516≥5

12.  P(z>0.219)=0.4133 Since p∡α, it is not statistically significant. Therefore we do not reject H₀. There is not enough evidence to say that the proportion of men who like shopping is greater than the proportion of women who like shopping for electronics.

13.  Example 3: We are interested in determining whether or not Spark notes is a reasonable substitute for actually reading a novel for English class. Many educators believe that students who rely on Spark notes do not do as well as those who read the novel and do not rely on Spark notes. Is there evidence that Spark notes users do not do as well as those who read the novel? Give a 90% confidence interval for the difference in proportions. A- or higherB+ or lower Reliant on spark notes 1915 Not reliant on spark notes 2910

14.  Two proportion z interval  Assumptions: -they’re independent -both random samples -Both populations are at least 10x sample -19≥1029 ≥10 15 ≥10 10≥10 -0.18475 +/- 0.18125=(-0.366,-0.0035) We are 90% confident that the difference of the true proportion of those who did good and did and did not use spark notes is between -0.366 and -0.0035.