Comparing Two Means Ch. 13

Two-Sample t Interval for a Difference Between Two Means

Degrees of Freedom Option 1: Use technology – decimals are possible. The calculator is using the formula Option 2: Compare the degrees of freedom of each sample (n 1 – 1) and (n 2 – 1). Use the smaller of the 2 groups. This is very conservative. Do #1 & 2

The null hypothesis has the general form H 0 : µ 1 - µ 2 = hypothesized value We’re often interested in situations in which the hypothesized difference is 0. Then the null hypothesis says that there is no difference between the two parameters: H 0 : µ 1 - µ 2 = 0 or, alternatively, H 0 : µ 1 = µ 2 The alternative hypothesis says what kind of difference we expect. H a : µ 1 - µ 2 > 0, H a : µ 1 - µ 2 < 0, or H a : µ 1 - µ 2 ≠ 0 Significance Tests for µ 1 – µ 2 If the Random, Normal, and Independent conditions are met, we can proceed with calculations.

Significance Tests for µ 1 – µ 2 To find the P-value, use the t distribution with degrees of freedom given by technology or by the conservative approach (df = smaller of n and n 2 - 1). Do #3 & 4

Robustness The 2 sample t procedures are more robust than the one- sample t methods When planning a two sample study, choose equal sample sizes if you can. When the shapes of the 2 populations are similar and the two samples are equal sizes, P – values are quite accurate. Two sample t procedures are most robust against non- Normality and the conservative P-values (using option 2 for df) are most accurate.

Using Two-Sample t Procedures Wisely In planning a two-sample study, choose equal sample sizes if you can. Do not use “pooled” two-sample t procedures! We are safe using two-sample t procedures for comparing two means in a randomized experiment. Do not use two-sample t procedures on paired data! Beware of making inferences in the absence of randomization. The results may not be generalized to the larger population of interest.

Pooling DON’T POOL!!!!! Pooling only occurs when the variances from both populations are equal. When does that every happen? Pooling was used before technology made it easy to calculate df for two samples.