Question

Question Help Test te dann about the dterence behieen two population means μ, and P2 at the level of s grificance α Assume the samples are 'andorn and independert and the po Ahns normally distributed Population statistics: σ' :3 4,02 : 1.6 Sample statistics 15,n 28,2-7, n2-27 Determine the alternative hypothesis. Determine the standardized test statistic z(Round to two decimal places as needed) Determine the P-value ae Round to three decimal places as needed) What is the proper decision? OA. Reject Ho There is not enough evdece at the 1% level of significance to repect the dam B. Fail to reed Ho OC. Fail to reject Ho There is enough evidence at the 1% level of significance to repect the dam OD, Reject Ho There is enough evidence at the 1% level o, sagnfcance to re ect the caím There is not enough evidence at the 1% level of sagntcance to reject the dam

Answer 1

Alternate Hypothesis :   $\mu_{1}\neq \mu_{2}$

It is known that :- $\bar{x_{1}}\sim N(\mu_{1},\sigma_{1}^2/n_{1})\;\;\;\;\;\;\;\;\bar{x_{2}}\sim N(\mu_{2},\sigma_{2}^2/n_{2})$

$\Rightarrow \bar{x_{1}}-\bar{x_{2}}\sim N(\mu_{1}-\mu_{2},\frac{\sigma_{1}^2}{n_{1}}+\frac{\sigma_{2}^2}{n_{2}})$

  $\Rightarrow \frac{\bar{x_{1}}-\bar{x_{2}}-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^2}{n_{1}}+\frac{\sigma_{2}^2}{n_{2}}}}\sim N(0,1)$

Under Null Hypothesis $\mu_{1}= \mu_{2}$ ; $\mu_{1}- \mu_{2}=0$

$\therefore$ Under Null Hypothesis ; $\frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{\sigma_{1}^2}{n_{1}}+\frac{\sigma_{2}^2}{n_{2}}}}\sim N(0,1)$

Thus the standardized test statistic is given by :-   

$Z= \frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{\sigma_{1}^2}{n_{1}}+\frac{\sigma_{2}^2}{n_{2}}}}$

Given :- $\sigma_{1}=3.4,\sigma_{2}=1.6,\bar{x_{1}}=15,\bar{x_{2}}=17,n_{1}=28,n_{2}=27$   

z = Observed value of Z

$z= \frac{15-17}{\sqrt{\frac{3.4^2}{28}+\frac{1.6^2}{27}}}=-2.37$

The test is a both-tailed test.

$\therefore$ P-value = 2 $\times$ min{ P(Z$\leqslant$z) , P(Z$\geqslant$z) } = 2 $\times$ min{ P(Z$\leqslant$-2.37) , P(Z$\geqslant$-2.37) } = 2 $\times$ min{0.009,0.991} = 2$\times$0.009 = 0.018

[We get the above probabilities from the Standard Normal table]

Level of significance $\alpha$ = 0.01

We see that ; p-value > $\alpha$

Thus the proper decision is :-

B. Fail to reject H₀​​​​​​​. There is not enough evidence at the 1% level of significance to reject the claim.