1. Section 2 Comparing two means
Chapter 11
Section 2
Comparing two means

2. 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.

3. 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.

4. 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.

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

6. 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

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

8. Table of values Population Sample size Sample Mean Sample sd 1 n1 x12 n2 x2 s2
This is how we describe the data, a table of values works best.

9. 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
Group 2 data – L2

10. 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

11. 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

12. 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

13. 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.

14. 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

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

16. 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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