## Confidence Intervals for the Mean

This section explains the need for confidence intervals and why a confidence interval is not the probability the interval contains the parameter. Then, it discusses how to compute a confidence interval on the mean when sigma is unknown and needs to be estimated. It also explains when to use t-distribution or a normal distribution. Next, it covers the difference between the shape of the t distribution and the normal distribution and how this difference is affected by degrees of freedom. Finally, it explains the procedure to compute a confidence interval on the difference between means.

### Difference between Means

#### Answers

- You assume that the values were sampled independently of each other, the populations have the same variance (homogeneity of variance), and the populations are normally distributed.
- df = (n1 - 1) + (n2 - 1) = (12 - 1) + (14 -1) = 24
- Difference between means = 3 - 2 = 1; MSE = (1 + 1.2)/2 = 1.1; Standard error = sqrt(2(1.1)/11) = .447; df = 20; t = 1.725; Lower limit = 1 - (1.725)(.447) = 0.229
- Enter this data into the Analysis Lab (or another statistical program). Your grouping variable (grade) should be composed of all 1's (1st graders) and 2's (2nd graders), and your dependent variable should contain the test scores. You should get the following confidence interval: (1.137, 5.263). If the statistical program calculates the mean of 1st - 2nd graders, you will have to reverse the confidence interval it gives you.