CHAPTER 11Section 11.2 – Comparing Two Means

COMPARING TWO MEANS

Comparing two populations or two treatments is one of the most common situations encountered in statistical practice. We call such situations two-sample problems.

A two sample problem can arise from a randomized comparative experiment that randomly divides subjects into two groups and exposes each group to a different treatment.

Comparing random samples separately selected from two populations is also a two sample problem. Unlike the matched pairs designs studied earlier, there is no matching of the units in the two samples and the two samples can be of different sizes.

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 11.9 - TWO-SAMPLE PROBLEMS

1. A medical researcher is interested in the effect on blood pressure of added calcium in our diet. She conducts a randomized comparative experiment in which one group of subjects receives a calcium supplement and a control group receives a placebo.

2. A psychologist develops a test that measures social insight. He compares the social insight of male college students with that of female college students by giving the test to a sample of students of each gender.

3. A bank wants to know which of two incentive plans will most increase the use of its credit cards. It offers each incentive to a random sample of credit card customers and compares the amount charged during the following six months.

CONDITIONS FOR COMPARING TWO MEANS

We have two SRSs, from two distinct populations.

Both populations are normally distributed. The means and standard deviations of the populations are unknown.

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.

ORGANIZING THE DATA

Call the variable we measure x1 in the first population and x2 in the second . We know parameters in this situation.

Population

Variable Mean Standard

deviation

1 x1 μ1 σ1

2 x2 μ2 σ2

ORGANIZING DATA (PART 2)

There are four unknown parameters, the two means and the two standard deviations.

Population

Sample Size

Sample Mean

Sample Standard deviation

1

2

THE SAMPLING DISTRIBUTION OF

The mean of is . That is, the difference of sample means is an unbiased estimator of the difference of population means.

The variance of the difference is the sum of the variance of which is

Note that the variances add. The standard deviations do not.

If the two populations are both normal, then the distribution of is also normal.

THE SAMPLING DISTRIBUTION OF (CONTINUED…)

Because the statistic has a normal distribution, we can standardize it to obtain a normal z statistic.

The two-sample z statistic is standardized by:

STANDARD ERROR

Because we don’t know the population standard deviations, we estimate them by the sample standard deviations from our two samples.

The result is the standard error, or estimated standard deviation:

The two-sample t statistic:

DEGREES OF FREEDOM FOR TWO-SAMPLE PROBLEMS

The two-sample t statistic does not have a t distribution since we replaced two standard deviations by the corresponding standard errors.

Therefore, we use two methods for calculating degrees of freedom:

Option 1: Use procedures based on the statistic t with critical values from a t distribution (used by calculator).

Option 2: Use procedures based on the statistic t with critical from the smaller n – 1.

TWO-SAMPLE T DISTRIBUTIONS

The statistic t has the same interpretation as any z or t statistic: it says how far is from its mean in standard deviation units.

When we replace just one standard deviation

in a z statistic by a standard error we must replace the z distribution with the t distribution.

CONFIDENCE INTERVAL FOR A TWO-SAMPLE T

The confidence interval for is given by:

is the upper (1-C)/2 critical value for the t(k) distribution with k the smaller of and

To test the hypothesis =, compute the two-sample t statistic:

Example: 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 were 21 healthy black men who volunteered to take part in the experiment. They were randomly assigned to two groups: 10 of the men received a calcium supplement for 12 weeks, while 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 Here are the data:

We want to perform a test of

H0: µ1 - µ2 = 0Ha: µ1 - µ2 > 0

where µ1 = the true mean decrease in systolic blood pressure for healthy black men like the ones in this study who take a calcium supplement, and µ2 = the true mean decrease in systolic blood pressure for healthy black men like the ones in this study who take a placebo.We will use α = 0.05.

Example: Calcium and Blood Pressure

If conditions are met, we will carry out a two-sample t test for µ1- µ2.

• Random The 21 subjects were randomly assigned to the two treatments.

• Normal With such small sample sizes, we need to examine the data to see if it’s reasonable to believe that the actual distributions of differences in blood pressure when taking calcium or placebo are Normal. Hand sketches of calculator boxplots and Normal probability plots for these data are below:

The boxplots show no clear evidence of skewness and no outliers. The Normal probability plot of the placebo group’s responses looks very linear, while the Normal probability plot of the calcium group’s responses shows some slight curvature. With no outliers or clear skewness, the t procedures should be pretty accurate.

• Independent Due to the random assignment, these two groups of men can be viewed as independent. Individual observations in each group should also be independent: knowing one subject’s change in blood pressure gives no information about another subject’s response.

Example: Calcium and Blood Pressure

Test statistic :

t (x 1 x 2) (1 2)

s12

n1

s22

n2

[5.000 ( 0.273)] 0

8.7432

10 5.901

2

11

1.604

Since the conditions are satisfied, we can perform a two-sample t test for the difference µ1 – µ2.

P-value Using the conservative df = 10 – 1 = 9, we can use Table C to show that the P-value is between 0.05 and 0.10.

Because the P-value is greater than α = 0.05, we fail to reject H0. The experiment provides some evidence that calcium reduces blood pressure, but the evidence is not convincing enough to conclude that calcium reduces blood pressure more than a placebo.

Example: Calcium and Blood Pressure

(x 1 x 2)t *s12

n1

s22

n2[5.000 ( 0.273)]1.833

8.7432

105.9012

11

5.2736.027

( 0.754,11.300)

We can estimate the difference in the true mean decrease in blood pressure forthe calcium and placebo treatments using a two-sample t interval for µ1 - µ2. To get results that are consistent with the one-tailed test at α = 0.05 from the example, we’ll use a 90% confidence level. The conditions for constructing a confidence interval are the same as the ones that we checked in the example before performing the two-sample t test.

With df = 9, the critical value for a 90% confidence interval is t* = 1.833.

The interval is:

We are 90% confident that the interval from -0.754 to 11.300 captures the difference in true mean blood pressure reduction on calcium over a placebo. Because the 90% confidence interval includes 0 as a plausible value for the difference, we cannot reject H0: µ1 - µ2 = 0 against the two-sided alternative at the α = 0.10 significance level or against the one-sided alternative at the α = 0.05 significance level.

THE POOLED TWO-SAMPLE PROCEDURES

Procedures that average use the statistical term “pooled.”

Pooled two-sample t procedures is a situation where the variances of both the samples are assumed to be the same and the sample sizes are the same. This rarely happens and the same results will occur with regular t procedures.

On the print out from a computer (p.666), use the unequal line for variances, degrees of freedom, and probability.

Homework: Technology toolbox on p.660-661 P.649 #’s 37, 38, 40, 43, & 50 Chapter 11 review: p.675 #’s 64, 68, & 72 Ch.10/11 take home test due Thursday 3/27

DO NOT WORK ON THE TEST TOGETHER!!!