Answer to Question #212306 in Statistics and Probability for Manasa Reddyrajula

Question

Consider the following data to validate the hypothesis that there is significant difference

between two populations.

Sample size

Mean

Standard deviation

20

25

5

20

20

6

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu_1=\mu_2$

$H_1:\mu_1\not=\mu_2$

This corresponds to a two-tailed test, and a z-test for two means, with known population standard deviations will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$ The rejection region for this two-tailed test is 

$R=\{z:|z|>1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x_1}-\bar{x_2}}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}$$

$$=\dfrac{25-20}{\sqrt{5^2/20+6^2/20}}=2.863$$

Since it is observed that $|z|=2.863>1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=2P(Z>2.863)=0.0042,$ and since $p=0.0042<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu_1$ is different than $\mu_2,$ at the $\alpha=0.05$ significance level.