Two samples are random and independent. Find the P-value used to test the claim that $\mu _ { 1 } \mu _ { 2 } . \text { Use } \alpha = 0.05 \text {. }$ Population statistics: $\sigma _ { 1 } = 40 \text { and } \sigma _ { 2 } = 24$ Sample statistics: $\overline { \mathrm { x } } 1 = 615 , \mathrm { n } _ { 1 } = 100 \text { and } \bar { x } _ { 2 } = 600 , \mathrm { n } _ { 2 } = 125$
A)0.0005
B)0.0505
C)0.1015
D)0.5105

Question

Two samples are random and independent. Find the P-value used to test the claim that $\mu _ { 1 } > \mu _ { 2 } . \text { Use } \alpha = 0.05 \text {. }$ Population statistics: $\sigma _ { 1 } = 40 \text { and } \sigma _ { 2 } = 24$ Sample statistics: $\overline { \mathrm { x } } 1 = 615 , \mathrm { n } _ { 1 } = 100 \text { and } \bar { x } _ { 2 } = 600 , \mathrm { n } _ { 2 } = 125$
A)0.0005
B)0.0505
C)0.1015
D)0.5105

Quizplus · Accepted Answer

To test the claim that $\mu_1 > \mu_2$, we use a Z-test for the difference between two means because the population standard deviations ($\sigma_1$ and $\sigma_2$) are known. The test statistic Z is calculated using the formula:$$ Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$Given $\bar{x}_1 = 615$, $\bar{x}_2 = 600$, $\sigma_1 = 40$, $\sigma_2 = 24$, $n_1 = 100$, and $n_2 = 125$, and assuming $\mu_1 - \mu_2 = 0$ under the null hypothesis, we find:$$ Z = \frac{(615 - 600) - 0}{\sqrt{\frac{40^2}{100} + \frac{24^2}{125}}} = \frac{15}{\sqrt{16 + 4.608}} = \frac{15}{\sqrt{20.608}} \approx \frac{15}{4.54} \approx 3.30$$Looking up the Z-score of 3.30 in the standard normal distribution table or using a calculator, we find that the area to the right (since we are testing for $\mu_1 > \mu_2$) corresponds to a very small P-value, which is closest to 0.0005, indicating strong evidence against the null hypothesis in favor of the alternative hypothesis that $\mu_1 > \mu_2$.

Two Samples Are Random and Independent μ1>μ2. Use α=0.05. \mu _ { 1 } > \mu _ { 2 } . \text { Use } \alpha = 0.05 \text {. }μ1​>μ2​. Use α=0.05.

Two Samples Are Random and Independent $\mu _ { 1 } > \mu _ { 2 } . \text { Use } \alpha = 0.05 \text {. }$