Section 11.2 Comparing Two Means AP Statistics

AP Statistics, Section Comparing Two Means Very useful to compare two populations Two populations equates to two distributions Easier math to work with one distribution Distribution of the difference of means

AP Statistics, Section 11.23

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5 Conditions 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 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 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 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 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.  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)

AP Statistics, Section 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 Inference Tool Box Step 3: Compute the test statistic and and the P- value.  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 Example Step 3: Compute the test statistic and the P-value. Remember the P- value is also dependent on the degrees of freedom.

AP Statistics, Section Exercises odd odd