o1 and o,Unknown Population means, Assumptions: independent Samples are randomly and samples independently drawn 01 and o2 known Populations are normally distributed or both sample sizes are at least 30 o,and o2 unknown Population variances are unknown but assumed equal Statistics for Managers Using Microsoft Excel Chap 9-16
Statistics for Managers Using Microsoft Excel Chap 9-16 Population means, independent samples σ1 and σ2 known σ1 and σ2 Unknown Assumptions: Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least 30 Population variances are unknown but assumed equal σ1 and σ2 unknown
o,and o,Unknown (continued) Population means, Forming interval estimates: independent samples The population variances are assumed equal,so use the two sample standard o1 and o2 known deviations and pool them to estimate o o,and o2 unknown the test statistic is a t value with (n1+n2-2)degrees of freedom Statistics for Managers Using Microsoft Excel Chap 9-17
Statistics for Managers Using Microsoft Excel Chap 9-17 Population means, independent samples σ1 and σ2 known σ1 and σ2 Unknown (continued) σ1 and σ2 unknown Forming interval estimates: The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ the test statistic is a t value with (n1 + n2 – 2) degrees of freedom
o1 and o,Unknown (continued) Population means, independent samples The pooled standard o1 and 02 known deviation is o,and o2 unknown S。 1-1s,2+2-1)S22 (n1-1)+(n2-1) Statistics for Managers Using Microsoft Excel Chap 9-18
Statistics for Managers Using Microsoft Excel Chap 9-18 Population means, independent samples σ1 and σ2 known σ1 and σ2 Unknown The pooled standard deviation is (continued) (n 1) (n 1) n 1S n 1S S 1 2 2 2 2 2 1 1 p σ1 and σ2 unknown
o1 and o,Unknown (continued) The test statistic for Population means, independent μ1-2iS: samples (区1-X2)-(山，-2) o and o2 known o,and 02 unknown Where t has (n+n2-2)d.f., and s-血-s,2+2-0s,2 (n1-1)+(n2-1) Statistics for Managers Using Microsoft Excel Chap 9-19
Statistics for Managers Using Microsoft Excel Chap 9-19 Population means, independent samples σ1 and σ2 known σ1 and σ2 Unknown Where t has (n1 + n2 – 2) d.f., and 1 2 2 p 1 2 1 2 n 1 n 1 S X X μ μ t The test statistic for μ1 – μ2 is: σ1 and σ2 unknown (n 1) (n 1) n 1 S n 1 S S 1 2 2 2 2 2 2 1 1 p (continued)
Confidence Interval, o1 and o,Unknown Population means, The confidence interval for independent samples μ1-2iS: o1 and o2 known o,and o2 unknown Where s3=血-s,+-s, (n1-1)+(n2-1) Statistics for Managers Using Microsoft Excel Chap 9-20
Statistics for Managers Using Microsoft Excel Chap 9-20 Population means, independent samples σ1 and σ2 known 1 2 2 n n -2 p 1 2 n1 n1 X X t S 1 2 The confidence interval for μ1 – μ2 is: Where (n 1) (n 1) n 1 S n 1 S S 1 2 2 2 2 2 2 1 1 p σ1 and σ2 unknown Confidence Interval, σ1 and σ2 Unknown