Question

Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations. (You may find it useful to reference the appropriate table: z table or t table) Ho: H1-Hu2 0 HA: H1 Hz< e 251 252 s1 39 s=19 n1=7 n 7 a-1. Calculate the value of the test statistic under the assumption that the population variances are equal. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.) Answer is complete and correct. -0.061 Test statistic

Answer 1

a-1. In case the equal population variance is assumed, the test statistic would be $t = \frac{\bar x_1 - \bar x_2}{\sqrt{(s_1^2 + s_2^2)/n}}$ , and for the given values, we have
251 252 (392+192) / 7
or
0.06098727-0.061
.

The degree of freedom is 2n-2 or 12.

a-2. The test would be a left tailed-test. The p-value would be , for x being the standard t-distribution with 12 df. As can be seen, the p-value is greater than 0.10.

P(rt) 0.4761817

a-3. As the p-value is higher than 0.01 alpha level, we fail to reject the null, and conclude that $\mu_1 - \mu_2 \geq 0$ or $\mu_1 \geq \mu_2$ . Hence, the correct option would be

• No, since the value of the p-value is greater than the significance level.

a-4. The correct option would be

• We cannot conclude that population mean 1 is less than population mean 2.

The reason being that we fail to reject the null which is $\mu_1 - \mu_2 \geq 0$ or $\mu_1 \geq \mu_2$ . The conclusion that population mean 1 is indeed less than population mean 2 would happen when the null hypothesis is rejected.

b-1. The test statistic assuming that population variances are unknown would be
t = /s1/n1+s/n2
, which would yield the same as before considering n1=n2. We have
251 252 /392/7192/ 7
or
0.06098727-0.061
.

b-2. The p-value would be the same as before, ie , which is greater than 0.10.

P(rt) 0.4761817

b-3. As before, the p-value is higher than 0.01 alpha level, we fail to reject the null, and conclude that $\mu_1 - \mu_2 \geq 0$ or $\mu_1 \geq \mu_2$ . Hence, the correct option would be

• No, since the value of the p-value is greater than the significance level.

b-4. The correct option would be

• We cannot conclude that population mean 1 is less than population mean 2.

The result and reasons remains same as before, since the test statistic and conclusion doesn't change. We again fail to reject the null, which is $\mu_1 \geq \mu_2$ .