1. SECTION 11.2 COMPARING TWO MEANS AP Statistics

2. Comparing Two Means AP Statistics, Section 11.2 2  Very useful compare two populations  Two population equates to two distributions, perhaps of different size  Easier math to work with one distribution  Distribution of the difference of means and one sample t-procedures when possible.

3. Some formulas: 3 AP Statistics, Section 11.2

4. Formulas (continued) AP Statistics, Section 11.2 4

5. 5

6. Assumptions for Comparing Two Means AP Statistics, Section 11.2 6  We have two SRSs, from two distinct populations.  The samples are independent. That is, one sample has no influence on the other. Matching violates independence, for example.  We measure the same variable for both samples.  Both populations are normally distributed. The means and standard deviations of the populations are unknown.

7. Example AP Statistics, Section 11.2 7  Does increasing the amount of calcium in our diet reduce blood pressure? Examination of a large sample of people revealed a relationship between calcium intake and blood pressure. The relationship was strongest for black men. Such observational studies do not establish causation. Researchers therefore designed a randomized comparative experiment.

8. Example AP Statistics, Section 11.2 8  The subjects in part of the experiment were 21 healthy black men. A randomly chosen group of 10 of the men received a calcium supplement for 12 weeks.  The group of 11 men received a placebo pill that looked identical.  The experiment was double-blind.  The response variable is the decrease in systolic (heart contracted) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative response.

9. Example AP Statistics, Section 11.2 9  Group 1 (Calcium) results:  7, -4, 18, 17, -3, -5, 1, 10, 11, -2  n=10, x-bar=5.000, s=8.743  Group 2 (Placebo) results:  -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3  n=11, x-bar=-0.273, s=5.901

10. Inference Tool Box AP Statistics, Section 11.2 10  Step 1: Identify the populations and the parameters of interest you want to draw conclusions about. State hypothesis in words and symbols.  Pop 1 : Black Men on Calcium;  Pop 2 : Black Men on Placebo  Parameters of interest: mean differences in blood pressure  H 0 : µ 1 = µ 2 (There is no difference in the blood pressure changes)  H a : µ 1 > µ 2 (The men taking calcium see a larger decrease in blood pressure)

11. Inference Tool Box AP Statistics, Section 11.2 11  Step 2: Choose the appropriate inference procedure, and verify the conditions for using the selected procedure.  Test? Because we don’t know the population standard deviation, we’ll use a t test. Since we’re not comparing a person with himself, we have two sample.  Independent? SRSs, therefore independent.  Normal? Use back-to-back stemplots to check for normality. The book says “no departures from normality”

12. Inference Tool Box AP Statistics, Section 11.2 12  Step 3: Compute the test statistic and and the P- value.

13. Notes on p-value  There are two options for calculating p-value:  Option 1: Use 2 sample t procedures from data and allow calculator to compute.  Option 2: Use procedures based on t-distribution with the smaller n to find d.f. AP Statistics, Section 11.2 13

14. Exercises AP Statistics, Section 11.2 14  11.2 HOMEWORK  11.33-11.35 all  11.37 – 11.39, 40, 43, 53, 54, 58, 62, 64  Due Friday, March27  Post Test Chapters 1 - 11 on Wed., April 1st