8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) Key Concepts: –Sampling Distribution of the Difference of the Sample.

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8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) Key Concepts: –Sampling Distribution of the Difference of the Sample Means –Pooled Estimate of the Standard Deviation –Two-Sample t-Test for the Difference Between Means

8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) In the last section, we assumed the variable of interest was normal (or approximately normal) on each population and that  1 and  2 were both known. In this section, we won’t have  1 or  2.  We need a new procedure.

8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) Review the sampling distribution of the difference of the sample means. –Since we will not have either population standard deviation, we need to adjust the standard error formula. If we can assume σ 1 = σ 2, we can pool the sample variances:

8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) When we pool the sample variances, the standard error of the difference of the sample means is: Our test statistic will be the studentized version of the difference of the sample means. this variable follows a t-distribution with df = n 1 + n 2 – 2.

8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) What happens if we cannot assume σ 1 = σ 2 ? –We are not allowed to pool the sample variances. –The next best option is to use the sample variances in place of the population variances in the original standard error formula:

8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) When we use this new studentized version of the difference of the sample means, we have: this variable follows a t-distribution with df equal to the smaller of n 1 – 1 or n 2 – 1. –The flowchart on page 429 summarizes what we’ve learned. –Guidelines for Two-Sample t-Tests are given on page 429.

8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown) Practice #12 p. 432 #14 p. 433 (Transactions) Pooled #20 p. 434 (Tensile Strength) Non-pooled