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

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.

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.

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.

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!

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 = 3924 s4 = 829.67 x8 = 4069.27 s8 = 952.90

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.

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 = 3924 s4 = 829.67 x8 = 4069.27 s8 = 952.90 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 -814.1 to 523.6 Kcal captures the true difference between the average food intake of sleeping 4 hours and the average food intake of sleeping 8 hours.

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