Comparing Two Means

Comparing Two Means Very useful compare two populations Two population equates to two distributions Easier math to work with one distribution Distribution of the difference of means AP Statistics, Section 11.2

AP Statistics, Section 11.2

AP Statistics, Section 11.2

AP Statistics, Section 11.2

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 11.2

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

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

Example Group 1 (Calcium) results: Group 2 (Placebo) 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

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

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

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 11.2

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 11.2

Comparing Two Proportions

Our inferential work so far… Has been about the distribution of sample means and the distribution of the difference of sample means. and the distribution of sample proportions

The sampling distribution of the difference of proportions

The shape of the sampling distribution of difference of proportions If these conditions are met, the sampling distribution of difference of proportions is approximately normal

Two proportion confidence interval

Example A study investigated the effect of pre-school on use of social services. What is the 95% confidence interval? Population Description Sample size Sample proportion 1 Control 61 .803 2 Preschool 62 .613

Calculate the Confidence Interval 1: Populations & Parameter of Interest 3: Calculations 2: Procedure Name & Conditions 4: Interpret

1: Population, Parameter of Interest, H0 and Ha 3: Calculations 2: Procedure Name & Conditions 4: Interpret

1: Population, Parameter of Interest, H0 and Ha 3: Calculations 2: Procedure Name & Conditions 4: Interpret