Question

Let X1,..,X40 be a random sample from a population with mean H1 and variance o 1-2.56, and let Y1, Y32 be a random sample from a population with mean 2 and variance of 2-1.96, and that Xand y samples are independent of one another. Assume the sample mean values are X-18 and y-17, and we want to test Ho: μ 1-μ 2-0 versus Ha: μ 1-u2 > 0, which of the following statements are correct? O Ho is rejected at the .05 level if Z s 1.65 Ho is rejected at the.05 level if z s 1.96 The value of the test statistic is Z 1.88 The value of the test statistic is Z-2.83

Answer 1

For sample 1:

$n_1= 40, \bar x = 18, \sigma_1^2= 2.56$

For sample 2:

$n_12= 32 , \bar y = 17, \sigma_2^2= 1.96$

The null and the alternative hypotheses are,

$H_0: \mu_1 - \mu_2 =0$

$H_a: \mu_1 - \mu_2 >0$

Test statistic is,

$Z = \frac { \bar x -\bar y}{\sqrt { \frac {\sigma_1^2}{n_1} + \frac {\sigma_2^2}{n_2}}}$

$Z = \frac {18 - 17}{\sqrt { \frac { 2.56}{40} + \frac {1.96 }{32}}}$

$Z = 2.83$

Since, it is right-tailed test, critical value at significance level $(\alpha)$ = 0.05 is, Z^* = 1.65

Decision rule: H₀ is rejected at the 0.05 level if Z >= 1.65

Here, test statistic = 2.83 > Z^* = 1.65

so, we reject the null hypothesis ( H₀).

Therefore, correct statement is,

The value of the test statistic is Z = 2.83