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Lecture No.2 Inference For Two Population Mean The pooled t test for two population means (critical-value approach) Assumptions 1.independent samples 2.normal populations or large samples 3.equal population standard deviations Step 1: the null hypothesis is H O : 1 2 and the alternative hypothesis is H a : 1 2 (two tailed ) H O : 1 2 (left tailed ) H O : 1 2 ( right tailed ) Step 2 : decide on the significance level, Step 3: compute the value of the test statistic T Where x1 x2 S p / (1 / n1 1 / n2 ) Sp (n1 1) s12 (n2 1) s22 n1 n2 2 Step 4: the critical value (s) are t / 2 (two tailed ) or t (left tailed ) or t (right tailed ) with degrees of freedom (df= n1 + n2 -2) Step 5 : if the value of the t test statistics falls in the rejection region, reject HO ; otherwise, fail to reject H0 Step 6 : interpret the results of the hypothesis test. The pooled t test for two population means (p-value approach) Assumptions 1.independent samples 2.normal populations or large samples 3.equal population standard deviations Step 1: the null hypothesis is H O : 1 2 and the alternative hypothesis is H a : 1 2 (two tailed ) H O : 1 2 (left tailed ) H O : 1 2 ( right tailed ) Step 2 : decide on the significance level, Step 3: compute the value of the test statistic T Where Sp x1 x2 S p / (1 / n1 1 / n2 ) (n1 1) s12 (n2 1) s22 n1 n2 2 Step 4: the value of t-statistics has df= n1 + n2 -2. Use a table to estimate the p-value or obtain it exactly by using technology. Step 5 : if the P- value less than or equal , ( p value ), reject HO ; otherwise, fail to reject H0 Step 6 : interpret the results of the hypothesis test. Example : we perform a hypotheses test to decide whether there is a difference between the mean salaries of faculty in public and private institutions. Independent random samples of 20 faculty members in public institutions and 35 faculty members in private institutions yield in the data in table below. At the 5% significance level, do the data provide sufficient evidence to conclude that means salaries for faculty in public and private institutions differ? Annual salary ($1000s)for 30 faculty members in public institutions and 35 faculty members in private institutions Sample 1 (public institutions) Sample 2 (private institutions) 34.2 56.8 58.2 29.2 60.2 92.9 62.9 45.2 66.3 47.2 71.0 90.0 41.4 76.8 15.8 88.2 52.0 53.8 76.0 31.1 59.3 97.3 100.4 35.0 84.2 33.8 44.6 63.1 101.0 56.1 71.1 97.5 92.6 24.6 54.2 79.4 40.2 64.4 118.5 68.6 77.6 73.5 27.2 56.0 37.7 51.5 61.6 67.6 81.2 62.3 56.0 81.8 41.2 71.0 102.2 46.4 78.3 52.4 24.8 107.4 24.4 42.2 51.2 74.0 63.6 Solution: Summary statistics for the samples public institutions private institutions x1 57.48 s1 23.95 n1 30 x2 66.39 s2 22.26 n2 35 Step 1: statethe null hypothesis and the alternative hypothesis H O : 1 2 ( mean salaries are the same) H a : 1 2 ( mean salaries are the different) Step 2 : decide on the significance level, 0.05 Step 3: compute the value of the test statistic T x1 x2 S p / (1 / n1 1 / n2 ) Where Sp T Sp (n1 1) s12 (n2 1) s22 n1 n2 2 (30 1)( 23.95) 2 (35 1) (22.26) 2 23.05 30 35 2 57.48 66.39 1.554 23.05 / (1 / 30 1 / 35) Critical-value approach Step 4: the critical value (s) are t / 2 (two tailed ) with degrees of freedom (df= n1 + n2 -2) From a table the critical values wit (df = 30+35-2=63) are t / 2 t0.05 / 2 1.998 Step 5 : if the value of the t test statistics falls in the rejection region, reject HO ; otherwise, fail to reject H0 From step 3 the value of the test statistics is t =-1.554, which does not fall in the rejection region, thus we do not reject HO . Step 6 : interpret the results of the hypothesis test. at 5% significance level, the data do not provide sufficient evidence to conclude that a difference exist between the mean salaries of faculty in public and private institutions . p-value approach Step 4: from a table(with df = 63) the p-value ( in two tailed) greater than 0.1 and less than 0.20 ( 0.1 < p < 0.2) , and by using technology, we obtain the p-value = 2 p ( t>= 1.554) = 0.125 (with df = 63) Step 5: p value < 0.05) so we reject HO at 5% significance level, the data do not provide sufficient evidence to conclude that a difference exist between the mean salaries of faculty in public and private institutions . use SPSS program use the SPSS program to perform the hypothesis in previous example STEP 1: Enter The Data As Shown Below Step 3 : the result shown below Tests of Normality a SALARY TYPE PRIV PUBL Kolmogorov-Smirnov Statistic df Sig. .080 35 .200* .105 30 .200* Shapiro-Wilk Statistic df .980 35 .975 30 *. This is a lower bound of the true significance. a. Lilliefors Significance Correction Group Statistics SALARY TYPE PUBL PRIV N 30 35 Mean 57.480 66.394 Std. Deviation 23.9528 22.2611 Std. Error Mean 4.3732 3.7628 Sig. .755 .680 Independent Sample s Te st SALARY Levene's Test for F Equality of Variances Sig. t-test for Equality of t Means df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Equal variances assumed .458 .501 -1.554 63 .125 Equal variances not assumed -8.914 -8.914 5.7363 5.7692 -20.3774 2.5488 -20.4549 2.6264 -1.545 59.853 .128