Section 2 Comparing two means Chapter 11 Section 2 Comparing two means

Two-sample problems The goal of inference is to compare the responses to two treatments or to compare the characteristics of two populations We have a separate sample from each treatment or each population.

Example of a two-sample problem A medical researcher is interested in the effect on blood pressure of added calcium in our diet. Conducts a randomized comparative experiment where one group of subjects receives a calcium supplement and the control group a placebo.

Example 2 A bank wants to know which of two incentive plans will most increase the use of its credit cards. Offers each incentive to a random sample and compares the amount charged during, say, six months.

Practice problem 11.38 11.38 (1)—single sample. (b) (3)—two samples.

Comparing two population means We can examine two-sample data graphically Stemplots (for small samples) Histograms Boxplots (for large samples) – remember the Hank Aaron and Barry Bonds comparison

Conditions for comparing two means We have two SRSs, from two distinct populations. The samples are INDEPENDENT. Both populations are normally distributed.

Table of values Population Sample size Sample Mean Sample sd 1 n1 x1 2 n2 x2 s2 This is how we describe the data, a table of values works best.

Calcium and Blood pressure Does increasing the amount of calcium in our diet reduce blood pressure? 21 healthy black men 10 men received calcium 11 men received placebo Group 1 data – L1 7 -4 18 17 -3 -5 1 10 11 -2 Group 2 data – L2 -1 12 -1 -3 3 -5 5 2 -11 -1 -3

Calculate a table of values Include n, x, and s Population Sample size Sample Mean Sample sd 1 – calcium 10 5.000 8.743 2 –Placebo 11 -0.273 5.901

Step 1 Identify the population of interest and the parameter you want to draw conclusions about Men taking calcium for 12 weeks Mean amount of calcium in diet

Null Hypothesis H0: µ1 = µ2 The mean decrease in blood pressure for those taking calcium is the same as the mean decrease in blood pressure for those taking a placebo

Alternative Hypothesis Ha: µ1 > µ2 The mean decrease in blood pressure for those taking calcium is greater than the mean decrease in blood pressure for those taking a placebo.

Step 2 Choose the appropriate inference procedure, and verify the conditions for using the selected procedure Two-sample t procedure Due to randomization, the two groups are independent SRSs Although the samples are small – check with a back-to-back stem and leaf

Step 3 Compute the test statistic And the P-value t = 1.604 P-value = 0.05 to 0.10

Step 4 Interpret your results in the CONTEXT of the PROBLEM The experiment found evidence that calcium reduces blood pressure, but the evidence falls short of the traditional 5% and 1% levels. We would fail to reject H0.

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