Summary statistics are given for independent simple random samples from two populations. Use the nonpooled t-test toconduct the required hypothesis test.
-$$\bar { x } _ { 1 } = 74.0 , \mathrm {~s} _ { 1 } = 4.5 , \mathrm { n } _ { 1 } = 11 , \overline { \mathrm { x } } _ { 2 } = 64.0 , \mathrm {~s} _ { 2 } = 5.1 , \mathrm { n } _ { 2 } = 9$$
Perform a two-tailed hypothesis test using a significance level of $\alpha = 0.01$.
A)Test statistic: $t = 4.598$
B) Test statistic: $t = 2.646$
Critical values $= \pm 2.878$
Critical values $= \pm 2.921$
Reject $\mathrm { H } _ { 0 }$
Do not reject $\mathrm { H } _ { 0 }$
C) Test statistic: $t = 2.646$
D) Test statistic: $t = 4.598$
Critical values $= \pm 2.878$
Critical values $= \pm 2.921$
Do not reject $\mathrm { H } _ { 0 }$
Reject $\mathrm { H } _ { 0 }$

Question

Summary statistics are given for independent simple random samples from two populations. Use the nonpooled t-test toconduct the required hypothesis test.
-$$\bar { x } _ { 1 } = 74.0 , \mathrm {~s} _ { 1 } = 4.5 , \mathrm { n } _ { 1 } = 11 , \overline { \mathrm { x } } _ { 2 } = 64.0 , \mathrm {~s} _ { 2 } = 5.1 , \mathrm { n } _ { 2 } = 9$$
Perform a two-tailed hypothesis test using a significance level of $\alpha = 0.01$.  
A)Test statistic: $t = 4.598$
B) Test statistic: $t = 2.646$
Critical values $= \pm 2.878$
Critical values $= \pm 2.921$
Reject $\mathrm { H } _ { 0 }$
Do not reject $\mathrm { H } _ { 0 }$
C) Test statistic: $t = 2.646$
D) Test statistic: $t = 4.598$
Critical values $= \pm 2.878$
Critical values $= \pm 2.921$
Do not reject $\mathrm { H } _ { 0 }$
Reject $\mathrm { H } _ { 0 }$

Quizplus · Accepted Answer

The test statistic is calculated as:$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} = \frac{74.0 - 64.0}{\sqrt{\frac{4.5^2}{11} + \frac{5.1^2}{9}}} = 4.598$$The critical values for a two-tailed test with a significance level of 0.01 and degrees of freedom equal to the smaller of $n_1-1$ and $n_2-1$ (which is 8 in this case) are $\pm 2.921$.Since the test statistic (4.598) is greater than the critical value (2.921), we reject the null hypothesis and conclude that there is a significant difference between the means of the two populations.

Summary Statistics Are Given for Independent Simple Random Samples from Two