Copyright © 2010 Pearson Education, Inc. Slide 24 - 1.

Copyright © 2010 Pearson Education, Inc. Slide 24 - 1


Solution: A

Copyright © 2010 Pearson Education, Inc.

Section 11.2

Comparing Two Means

Slide 24 - 3


Plot the Data

The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:


Comparing Two Means

Once we have examined the side-by-side boxplots, we can turn to the comparison of two means.

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

The standard deviation of the difference between two sample means is

We still don’t know the true standard deviations of the two groups, so we need to estimate and use the standard error

2 21 2

1 21 2

SD y yn n

2 21 2

1 21 2

s sSE y y

n n


Comparing Two Means (cont.)

Because we are working with means and estimating the standard error of their difference using the data, we shouldn’t be surprised that the sampling model is a Student’s t. The confidence interval we build is called a

two-sample t-interval (for the difference in means).

The corresponding hypothesis test is called a two-sample t-test.


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.

We estimate the standard error with

1 2 1 2

1 2

y yt

SE y y

2 21 2

1 21 2

s sSE y y

n n


Assumptions and Conditions

Independence Assumption (Each condition needs to be checked for both groups.): Randomization Condition: Were the data

collected with suitable randomization (representative random samples or a randomized experiment)?

10% Condition: We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.


Assumptions and Conditions (cont.)

Normal Population Assumption: Nearly Normal Condition: This must be

checked for both groups. A violation by either one violates the condition.

Independent Groups Assumption: The two groups we are comparing must be independent of each other. (See Chapter 25 if the groups are not independent of one another…)


TI – Tips Confidence Interval and hypothesis tests for difference in means (from two independent samples) For confidence intervals …

STAT TESTS 0:2-SampTint For hypothesis testing …

STAT TESTS 4:2-sampleTTest


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, 1 – 2.The confidence interval is

where the standard error of the difference of the means is

The critical value t*df 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.

2 21 2

1 21 2

s sSE y y

n n

1 2 1 2dfy y t SE y y


Degrees of Freedom

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

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

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


Testing the Difference Between Two Means

The hypothesis test we use is the two-sample t-test for 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.


A Test for the Difference Between Two Means

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

The standard error is

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.

We NEVER pool with a two-sample t-test for means.

1 2 0

1 2

y yt

SE y y

2 21 2

1 21 2

s sSE y y

n n


Example: Can you tell how much you are eating from how full you are? Or do you need visual cues? Researchers constructed a table with two ordinary 18 oz soup bowls and two identical looking bowls that had been modified to slowly, imperceptibly, refill as they were emptied. They assigned experiment participants to the bowls randomly and served them tomato soup. Those eating from the ordinary bowls had their bowls refilled by ladle whenever they were one-quarter full. If people judged their portions by internal cues, they should eat about the same amount.

Ordinary bowl n = 27, y bar = 8.5 oz. s = 6.1 oz. Refilling bowl n = 27, y bar = 14.7 oz. s = 8.4 oz.How much variability do we expect in the difference between

the two means?


Example: Researchers randomly assigned people to eat soup from one of two bowls: 27 got ordinary bowls that were refilled by ladle and 27 others were given bowls that secretly refilled as the people ate. The histograms for both groups look unimodal but somewhat skewed to the right. Can the researchers use their data to make inferences about the role of visual cues in determining how much people eat?


Example: What does a 95% confidence interval say about the difference in mean amounts eaten?


Example: Many office “coffee stations” collect voluntary payments for the food consumed. Researchers at the Univ. of Newcastle performed an experiment to see whether the image of eyes watching would change employee behavior. They alternated pictures of eyes looking at the viewer with pictures of flowers each week on the cupboard behind the “honesty box”. They measured the consumption of milk to approximate the amount of food consumed and recorded the contributions each week per liter of milk.

Eyes n = 5 weeks, ybar = $.417 per liter, s = $.1811 per liter

Flowers n = 5 weeks, ybar = $.1811 per liter, s =$0.67 per liter

Do these results provide evidence that there really is a difference in honesty even when it’s only photographs of eyes that are watching?

Copyright © 2010 Pearson Education, Inc. Slide 24 - 1.

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