Question

4. [7 marks] Consider two independent populations. We took two SRS of sizes 10 from each population and X1 = 525.751, sı 107.121, X2 = 373.269 and s2 = 67.498 Is there enough evidence to reject the null hypothesis in the following test? (a = 1%) Ho: H1 - H2 = 0 Ha:Mi - H2> 0

Answer 1

Answers : At 1% level of significance there is enough evidence to reject the null hypothes.

Forgot to mention degress of freedom.

The degrees of freedom in this two sample test hypothesis= n1+n2-2 = 10+10-2 = 18

df= 18

The detailed solution is given in the attached file. Hope this helps.

The provided sample means are shown below: X1 = 525.751 X2 = 373.269 Also, the provided sample standard deviations are: s] = 107.121 82 = 67.498 and the sample sizes are ni 10 and n2 = 10. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Mi = M2 Ha: ui > 2 This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.01, and the degrees of freedom are df = 18. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal: Hence, it is found that the critical value for this right-tailed test is te = 2.552, for a = 0.01 and df = 18. The rejection region for this right-tailed test is R = {t:t > 2.552}.

(3) Test Statistics Since it is assumed that the population variances are equal, the t-statistic is computed as follows: t= X1 - X2 (n1-1){+(m2-1):3 (1 + ) 525.751 - 373.269 (10-1)107.1212 +(10-1)67.4982 10+10-2 10 = 3.808 10 (4) Decision about the null hypothesis Since it is observed that t = 3.808 > tc = 2.552, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p = 0.0006, and since p= 0.0006 <0.01, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean ui is greater than 42, at the 0.01 significance level.