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

532 Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two VariancesHypothesis-Testing Summary 11. Comparison of a sample mean with a specificpopulation mean.Example: H 0 : m 100a. Use the z test when s is known:z X s m2 nb. Use the t test when s is unknown:t X m with d.f. n 1s2n2. Comparison of a sample variance or standard deviationwith a specific population variance or standard deviation.Example: H 0 : s 2 225Use the chi-square test:n x 2 1s2 with d.f. n 1s 23. Comparison of two sample means.Example: H 0 : m 1 m 2a. Use the z test when the population variances areknown:z X 1 X 2 m 1 m 2 s 2 1 s2 2An 1b. Use the t test for independent samples when thepopulation variances are unknown and assumethe sample variances are unequal:t X 1 X 2 m 1 m 2 s 2 1 s2 2An 1n 2n 2with d.f. the smaller of n 1 1 or n 2 1.Formula for the t test for comparing two means(independent samples, variances equal):Xt 1 X 2 m 1 m 2 n 1 1s 2 1 n 2 1s 2 2 1 1 A n 1 n 2 2 An 1 n 2with d.f. n 1 n 2 2.c. Use the t test for means for dependent samples:Example: H 0 : m D 0t D m Ds D 2nwhere n number of pairs.4. Comparison of a sample proportion with a specificpopulation proportion.Example: H 0 : p 0.32Use the z test:z X ms5. Comparison of two sample proportions.Example: H 0 : p 1 p 2Use the z test:z pˆ 1 pˆ 2 p 1 p 2 wherep X 1 X 2n 1 n 2pˆ 1 X 1n 1q 1 p pˆ 2 X 2n 26. Comparison of two sample variances or standarddeviations.Example: H 0 : s 2 1 s 2 2Use the F test:F s2 1s 2 2whereorA p q ¢ 1 n 1 1 n 2≤with d.f. n 1z pˆ p2pqns 2 1 larger variance d.f.N. n 1 1s 2 2 smaller variance d.f.D. n 2 19–62