Bluman A.G. Elementary Statistics- A Step By Step Approach

486 Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two VariancesConfidence intervals can also be found for the difference between two means withthis formula:Confidence Intervals for the Difference of Two Means:Independent SamplesVariances assumed to be unequal:X 1 X 2 t a 2As 2 1 s2 2 mn 1 n 1 m 2 X 1 X 2 t a2 s2 22 An 1 n 2d.f. smaller value of n 1 1 or n 2 1s 2 1Example 9–5 Find the 95% confidence interval for the data in Example 9–4.SolutionSubstitute in the formula.X 1 X 2 t a 2 As 2 1n 1 s2 2n 2 m 1 m 2191 199 2.365 A38 241.02 m 1 m 2 25.02s X 1 X 2 t 2 1a 2 s2 2An 1 n 28 12210 m 1 m 2 191 199 2.365 A38 28 12210Since 0 is contained in the interval, the decision is to not reject the null hypothesisH 0 : m 1 m 2 .9–16In many statistical software packages, a different method is used to compute thedegrees of freedom for this t test. They are determined by the formulasd.f. 2 1n 1 s 2 2n 2 2s 2 1n 1 2 n 1 1 s 2 2n 2 2 n 2 1This formula will not be used in this textbook.There are actually two different options for the use of t tests. One option is used whenthe variances of the populations are not equal, and the other option is used when the variancesare equal. To determine whether two sample variances are equal, the researchercan use an F test, as shown in Section 9–5.When the variances are assumed to be equal, this formula is used andX 1 X 2 m 1 m 2 t n 1 1s 2 1 n 2 1s 2 2A n 1 n 2 21 1 An 1 n 2follows the format ofobserved value expected valueTest value standard errorFor the numerator, the terms are the same as in the previously given formula. However,a note of explanation is needed for the denominator of the second test statistic. Since both