Chapter 9 Testing the Difference Between Two Means

Chapter 9 Testing the Difference Between Two Means T-test

What Will I Learn in 9. 5? Objectives: • Test the difference between two means for independent samples using the t test

9. 5 Testing the Difference Between Two Means: Using the t Test Formula for the t test for comparing two means from independent populations with unequal variances where the degrees of freedom are equal to the smaller of n 1 – 1 or n 2 – 1.

Assumptions are similar to Hypothesis Testing – Differences Between Means (z-test) Except for an additional assumption regarding variance. Assumptions: 1. The samples must be independent of each other. That is, there can be no relationship between the subjects in each sample. 2. If the sample sizes are less than 30, the populations must be normally or approximately normally distributed. 3. The level of measurement is interval-ratio 4. The variances of the populations are not equal.

Example: Farm Sizes The average size of a farm in Indiana County, Pennsylvania, is 191 acres. The average size of a farm in Greene County, Pennsylvania, is 199 acres. Assume the data were obtained from two samples with standard deviations of 38 and 12 acres, respectively, and sample sizes of 8 and 10, respectively. Can it be concluded at α = 0. 05 that the average size of the farms in the two counties is different? Assume the populations are normally distributed. Step 0 – Assumptions: Step 1: State the hypotheses and identify the claim. H 0: μ 1 = μ 2 and H 1: μ 1 μ 2 (claim)

Example : Farm Sizes Step 2: Find the critical values. Since the test is two-tailed, a = 0. 05, the degrees of freedom are the smaller of n 1 – 1 or n 2 – 1. In this case, the degrees of freedom are 8 – 1 = 7. The critical values are -2. 365 and 2. 365. Step 3: Find the test value (by hand).

Step 3: Find the test value (using technology). Geo. Gebra can be used to calculate the t-test value. 7

Example: Farm Sizes Step 4: Make the decision. Do not reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that the average size of the farms is different.

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A company collects data on the length of lunch breaks (in minutes) made by employees in two different departments: sales and shipping/receiving. The mean and standard deviation for the for a sample of 57 individuals in the sales division are 30. 26 and 1. 75 minutes respectively. The mean and standard deviation for a sample of 60 individuals in the shipping/receiving department is 25. 3 and 7. 5 minutes respectively. Use a significance level of 0. 05 to test the claim that the sales team takes longer lunch breaks.