Question #212306

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



1
Expert's answer
2021-07-01T11:00:27-0400

The following null and alternative hypotheses need to be tested:

H0:μ1=μ2H_0:\mu_1=\mu_2

H1:μ1μ2H_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 α=0.05,\alpha=0.05, and the critical value for a two-tailed test is zc=1.96.z_c=1.96. The rejection region for this two-tailed test is 

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

The z-statistic is computed as follows:


z=x1ˉx2ˉσ12/n1+σ22/n2z=\dfrac{\bar{x_1}-\bar{x_2}}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}

=252052/20+62/20=2.863=\dfrac{25-20}{\sqrt{5^2/20+6^2/20}}=2.863

Since it is observed that z=2.863>1.96=zc,|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,p=2P(Z>2.863)=0.0042, and since p=0.0042<0.05=α,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 μ1\mu_1 is different than μ2,\mu_2, at the α=0.05\alpha=0.05 significance level.



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