1. Comparing Two Populations or Treatments
Chapter 11 Lesson 11.1a
Comparing Two Populations or Treatments
11.1: Inferences Concerning the Difference Between 2 Population or Treatment Means Using Independent Samples

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

3. Assumptions and Conditions
Each condition needs to be checked for both groups
1. Random Condition
2. Normal Condition: Either both sample sizes are > 30 so the CLT applies or both the population distributions are approximately normal
3. (*NEW!*) Independent Groups Assumption: The two groups we are comparing must be independent of each other.

4. Two-Sample t-Interval
When the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups.
The confidence interval is
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.

5. Degrees of Freedom
The special formula for the degrees of freedom for our t critical value is a monster:
Because of this, we will let technology calculate degrees of freedom for us!

6. Find the mean and standard deviation for each treatment.
In a study on food intake after sleep deprivation, men were randomly assigned to one of two treatment groups. The experimental group were required to sleep only 4 hours on each of two nights, while the control group were required to sleep 8 hours on each of two nights. The amount of food intake (Kcal) on the day following the two nights of sleep was measured. Compute a 95% confidence interval for the true difference in the mean food intake for the two sleeping conditions.
4-hour sleep
3585
4470
3068
5338
2221
4791
4435
3099
3187
3901
3868
3869
4878
3632
4518
8-hour sleep
4965
3918
1987
4993
5220
3653
3510
3338
4100
5792
4547
3319
3336
4304
4057
Find the mean and standard deviation for each treatment.
x4 = s4 = x8 = s8 =

7. Food Intake Study Continued . . .
Verify the conditions
Conditions:
Men were randomly assigned to two treatment groups
2) The assumption of normality is plausible because both samples are fairly normal.
4000
4-hour
8-hour
3) It is fair to assume the groups are independent.

8. Calculate the interval.
Food Intake Study Continued . . .
4-hour sleep
3585
4470
3068
5338
2221
4791
4435
3099
3187
3901
3868
3869
4878
3632
4518
8-hour sleep
4965
3918
1987
4993
5220
3653
3510
3338
4100
5792
4547
3319
3336
4304
4057
x4 = s4 = x8 = s8 =
Based upon this interval, is there a significant difference in the mean food intake for the two sleeping conditions?
No, since 0 is in the confidence interval, there is not convincing evidence that the mean food intake for the two sleep conditions are different.
Calculate the interval.
We are 95% confident that the interval to Kcal captures the true difference between the average food intake of sleeping 4 hours and the average food intake of sleeping 8 hours.

9. Homework
Pg.549: #12, (17c), 28a