Section 11.2 Comparing Two Means

1. Section 11.2Comparing Two Means AP Statistics

2. Comparing Two Means • 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. AP Statistics, Section 11.2

3. Some formulas: AP Statistics, Section 11.2

4. Formulas (continued) AP Statistics, Section 11.2

5. AP Statistics, Section 11.2

6. Assumptions for Comparing Two Means • 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. AP Statistics, Section 11.2

7. Example • 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. AP Statistics, Section 11.2

8. Example • 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. AP Statistics, Section 11.2

9. Example • 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 AP Statistics, Section 11.2

10. Inference Tool Box • Step 1: Identify the populations and the parameters of interest you want to draw conclusions about. State hypothesis in words and symbols. • Pop1: Black Men on Calcium; • Pop2: Black Men on Placebo • Parameters of interest: mean differences in blood pressure • H0: µ1= µ2 (There is no difference in the blood pressure changes) • Ha: µ1> µ2 (The men taking calcium see a larger decrease in blood pressure) AP Statistics, Section 11.2

11. Inference Tool Box • 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” AP Statistics, Section 11.2

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

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

14. Exercises • 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 AP Statistics, Section 11.2