Chapter 24

Comparing Means

Comparing two means is not very different from comparing two proportions.

This time the parameter of interest is the difference between the two means, 1 – 2.

So, the standard deviation of the difference between two sample means is

Since we don’t know the true standard deviations, we need to use the standard errors

Comparing Two Means

2 21 2

1 21 2

SD y yn n

2 21 2

1 21 2

s sSE y y

n n

Independence Assumption (Each condition needs to be checked for both groups.):◦ Randomization Condition:◦ 10% Condition:◦ Nearly Normal Condition:

Independent Groups Assumption

Assumptions and Conditions

2 21 2

1 21 2

s sSE y y

n n

Two-Sample t-IntervalWhen the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups:

where the standard error of the difference of the means is

The critical value depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, which we get from the sample sizes and a special formula.

1 2 1 2dfy y t SE y y

The special formula for the degrees of freedom for our t critical value is:

Because of this, we will let technology calculate degrees of freedom for us!

Degrees of Freedom

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

s sn n

dfs s

n n n n

A researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The data she collected are shown below:

Create a 90% confidence interval for the difference in mean pulse rates.

Example

Male Female

Count 28 24

Mean 72.75 72.625

Median 73 73

StdDev 5.37225 7.69987

Range 20 29

IQR 9 12.5

1 2 1 2

1 2

y yt

SE y y

Sampling Distribution for the Difference Between Two Means

When the conditions are met, the standardized sample difference between the means of two independent groups

can be modeled by a Student’s t-model with a number of degrees of freedom found with a special formula.

The hypothesis test we use is the 2-sample t-test for the difference between means.

The conditions for the two-sample t-test for the difference between the means of two independent groups are the same as for the two-sample t-interval.

We test the hypothesis H0: 1 – 2 = 0, where the hypothesized difference, 0, is almost always 0, using the statistic

When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s t-model with a number of degrees of freedom given by a special formula. We use that model to obtain a P-value.

Testing the Difference Between Two Means

1 2 0

1 2

y yt

SE y y

Someone claims that the generic batteries seem to have last longer than the brand-name batteries. To prove this claim, we measured the lifetimes of 6 sets of generic and 6 sets of brand-name AA batteries from a randomized experiment.

Does this prove the claim?

Example

Generic Brand-name

Count 6 6

Mean 206 min 187.4 min

StdDev 10.3 min 14.6 min

Page 640 – 643 Problem # 1, 3ab, 11, 13, 15, 17, 25, 31.

Homework Assignment