Chapter 9 Inferences from Two Samples Lecture 1 Sections: 9.1 – 9.2

There are many real and important situations in which it is necessary to use sample data to compare two population proportions. We may want to compare the percentage of males and females in higher education. Also, we might want to determine whether there is a difference between the percentage of gas consumption in two different automobile types. When testing a hypothesis about two population proportions or when constructing a confidence interval for the difference between two population proportions, we have to make the following assumptions and use the following notation. Assumptions: 1.We have proportions from two simple random samples that are independent. Sample values selected from one population are not related. 2. For both samples, the conditions np ≥ 5 and nq ≥ 5 are satisfied. That is, there are at least five successes and five failures in each of the two samples.

Notation for Two Proportions:

Confidence Intervals Estimate for p 1 –p 2 : Recall that when you are trying to determine a confidence interval or performing a Hypothesis about a proportion, you will always use Z

1. A personal trainer randomly selected 400 females, and 340 weight train. In 450 randomly selected males, the trainer found that 406 weight train. Construct a 95% confidence interval for (p male – p female ). 2. A random sample of 300 male students at ELAC shows that 230 of them have internet access at home while a random sample of 350 female students shows that 290 of them have internet access at home. Find the 99% confidence interval for the difference of the two proportions.

3. The county mental health department claims that a smaller proportion of the crimes committed by persons younger than 21 years of age are violent crimes when compared to the crimes committed by persons 21 years of age or older. Of 2750 randomly selected arrests of criminals younger than 21 years of age, 4% involve violent crimes. Of 2200 randomly selected arrests of criminals 21 years of age or older, 4.5% involve violent crimes. Construct a 98% confidence interval for the difference between the two proportions of violent crimes. Minitab Printout: Test and CI for Two Proportions Sample X N Sample p Difference = p (1) - p (2) Estimate for difference: % CI for difference: ( , ) Test for difference = 0 (vs not = 0): Z = P-Value = 0.388

Hypothesis Testing About 2 Proportions: Test Statistic for H 0 : p 1 = p 2 ↔ p 1 – p 2 = 0

5. In s study done in Miami’s economically disadvantaged community, it was determined that 38 out of 62 children who attended preschool needed social services later in life compared to 49 out of 61 children who did not attend preschool. Test the claim that the proportion of children who need social services later in life is greater amongst the children who did not attend preschool compare to the ones who did attend preschool. 4. A random sample of 300 male students at ELAC shows that 230 of them have internet access at home while a random sample of 350 female students shows that 290 of them have internet access at home. Test the claim that the percentage of males who have internet access is larger than that of females at a 0.05 level of significance.

5. In a Time magazine survey, 24% of 200 single women said that they “definitely want to get married.” In the same survey, 26% of 250 single men gave the same response. Test the claim that the percentage of females who “definitely want to get married” is less than that of males at a 0.01 level of significance. Minitab Output: Test and CI for Two Proportions Sample X N Sample p Difference = p (1) - p (2) Estimate for difference: % upper bound for difference: Test for difference = 0 (vs < 0): Z = P-Value = 0.313