Comparing Two Comparing Two Population ParametersPopulation ParametersComparing 2 MeansComparing 2 Means

Sample HT’s for 2-Sample HT’s for 2-sample Problemssample Problems

Which of 2 popular drugs - Lipitor Which of 2 popular drugs - Lipitor or Prevachol - helps lower “bad or Prevachol - helps lower “bad cholesterol?”cholesterol?”

Who is more likely to binge drink: Who is more likely to binge drink: male or female college students?male or female college students?

Which of the 2 incentive plans Which of the 2 incentive plans from a bank will increase the use from a bank will increase the use of credit cards?of credit cards?

Conditions for Conditions for comparing 2 “Means”comparing 2 “Means”

• SRS We have two SRSs, from two distinct populations.

• Normality Both populations are Normally distributed. In practice, it is enough that the distributions have similar shapes and that the data have no strong outliers.

• Independence The samples are independent. That is, one sample has no influence on the other. When sampling without replacement from two distinct populations, use Rule of Thumb 1 (in symbols, N ≥ 10n).

Two-sample z statistic

Test-Statistic

Two-sample t statisticBecause we don't know the population standard

deviations, we estimate them by the standard deviations from our two samples. The result is the standard error, or estimated standard deviation, of the difference in sample means:

Test-Statistic

Today: 2-Sample Problems

•Hypothesis Testing

•Confidence Interval

Case 1: calcium and blood pressure 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.

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 control group of 11 men received a placebo pill that looked identical. The experiment was double-blind. The response variable is the decrease in systolic (top number) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative response.

Take Group 1 to be the calcium group and Group 2 the placebo group. Here are the data for the 10 men in Group 1 (calcium),

and for the 11 men in Group 2 (placebo),

From the data, calculate the summary statistics:

Step 1: Hypotheses We write hypotheses in terms of the mean decreases we would see in the entire population μ1 for men taking calcium for 12 weeks and μ2 for men taking a placebo. The hypotheses are:

Step 2: Conditions:

SRS The 21 subjects in this experiment were not obtained by random selection from a larger population. However, the random assignment of subjects to treatments should help ensure random sampling.

Normality Inspection of the data reveals that there are no outliers.

The placebo responses appear roughly Normal.

Independence Because of the randomization, we are willing to regard the calcium and placebo groups as two independent samples. We are not sampling without replacement from a population of interest in this case.

Step 2: Conditions: