1. Comparing and Testing two proportions 12.3

2. Breast feeding mothers secrete calcium into their milk. Some of the calcium may come from their bones, so mothers many lose bone mineral. Researchers compared 47 breast-feeding women with 22 women of similar age who were neither pregnant nor lactating. They measured the percent change in mineral content of the women’s spines over three months. Here are the data. Do these data give good evidence that on the average nursing mothers lose more bone mineral? Give appropriate statistical evidence to support your response.

3. A growing number of employers are trying to hold down the cost that they pay for medicine insurance for their employees. As part of this effort, many require clients to use generic brand medicines when filling prescriptions. An independent consumer advocacy group wanted to determine if there was a difference in milligrams, in the amount of active ingredient between a certain “name” brand drug and its generic counterpart. Pharmacies may store drugs under different conditions. Therefore, the consumer group randomly selected ten different pharmacies in a large city and filled two prescriptions at each of those pharmacies, one for the “name” brand drug and the other for the generic brand of the drug. The consumers group’s laboratory then tested a randomly selected pill from each prescription to determine the amount of active ingredient in the pill. The results are given in the following table.

4. Some Review: We need to begin by reviewing some facts that we learned earlier: the combination of two samples and what happened to the mean and standard deviation and then we need to apply this knowledge to two proportions!!! If I combine sets A and B then μ A + B =  A +  B σ A + B =

5. More Review: μ A - B =  A –  B σ A - B =

6. Apply this to proportions So proportions p A + B = p A + p B and the σ(p) A + B = And p A - B = p A - p B and the σ(p) A - B = This leads us to believe that if we form a sampling distribution for a difference between two independent proportions we get:

7. Sampling Distribution Model for the difference between two Independent Proportions If the samples are independent and the sampled values are independent and the sample sizes are large enough the sampling distributions of is approximately Normal with  = and

8. We must satisfy the assumptions: Both are SRS Both Events are independent (10n<Population) Both are Approximately Normal: We can now run two proportion z-intervals and two proportion z-tests.

9. Two Proportion z- interval Remember the formula we used for 1-proportion z-intervals? Use the previous facts to change the formula for two proportions:

10. Among 242 Cleveland area children born prematurely at low birth weights between 1977 and 1979, only 74% graduated from high school. Among a comparison group of 233 children of normal birth weight, 83% were high school graduates. Create a 95% confidence interval for the difference in graduation rates between children of normal and very low birth weights. Does this provided evidence that premature birth may be a risk factor for not finishing high school? Use your confidence interval to test an appropriate hypothesis. Suppose your conclusion is incorrect. Which type of error did you make?

11. Now we will apply your knowledge to hypothesis testing. The first thing we need to do is to decide how to write the null and alternative hypothesis. The new null hypothesis always becomes H o : p 1 – p 2 = 0 (the null is always that there is no difference) and the alternative will be either: Note: It is very important to write a let statement indicating which way you are subtracting!!!

12. Example: There is some indication in medical literature that doctors become more aggressive in inducing labor or doing preterm caesarean sections when a woman is carrying twins. Records at a large hospital show that of the 43 sets of twins born in 1990, 20 were delivered before the 37 th week of pregnancy. In 2000, 26 of 48 sets of twins were born preterm. Does this indicate and increase in the incidence of early births in twins? Test an hypothesis and state our conclusion.