Answer:
[tex](a)\ \bar x_1 - \bar x_2 = 2.0[/tex]
[tex](b)\ CI =(1.0542,2.9458)[/tex]
[tex](c)\ CI = (0.8730,2.1270)[/tex]
Step-by-step explanation:
Given
[tex]n_1 = 60[/tex] [tex]n_2 = 35[/tex]
[tex]\bar x_1 = 13.6[/tex] [tex]\bar x_2 = 11.6[/tex]
[tex]\sigma_1 = 2.1[/tex] [tex]\sigma_2 = 3[/tex]
Solving (a): Point estimate of difference of mean
This is calculated as: [tex]\bar x_1 - \bar x_2[/tex]
[tex]\bar x_1 - \bar x_2 = 13.6 - 11.6[/tex]
[tex]\bar x_1 - \bar x_2 = 2.0[/tex]
Solving (b): 90% confidence interval
We have:
[tex]c = 90\%[/tex]
[tex]c = 0.90[/tex]
Confidence level is: [tex]1 - \alpha[/tex]
[tex]1 - \alpha = c[/tex]
[tex]1 - \alpha = 0.90[/tex]
[tex]\alpha = 0.10[/tex]
Calculate [tex]z_{\alpha/2}[/tex]
[tex]z_{\alpha/2} = z_{0.10/2}[/tex]
[tex]z_{\alpha/2} = z_{0.05}[/tex]
The z score is:
[tex]z_{\alpha/2} = z_{0.05} =1.645[/tex]
The endpoints of the confidence level is:
[tex](\bar x_1 - \bar x_2) \± z_{\alpha/2} * \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}[/tex]
[tex]2.0 \± 1.645 * \sqrt{\frac{2.1^2}{60}+\frac{3^2}{35}}[/tex]
[tex]2.0 \± 1.645 * \sqrt{\frac{4.41}{60}+\frac{9}{35}}[/tex]
[tex]2.0 \± 1.645 * \sqrt{0.0735+0.2571}[/tex]
[tex]2.0 \± 1.645 * \sqrt{0.3306}[/tex]
[tex]2.0 \± 0.9458[/tex]
Split
[tex](2.0 - 0.9458) \to (2.0 + 0.9458)[/tex]
[tex](1.0542) \to (2.9458)[/tex]
Hence, the 90% confidence interval is:
[tex]CI =(1.0542,2.9458)[/tex]
Solving (c): 95% confidence interval
We have:
[tex]c = 95\%[/tex]
[tex]c = 0.95[/tex]
Confidence level is: [tex]1 - \alpha[/tex]
[tex]1 - \alpha = c[/tex]
[tex]1 - \alpha = 0.95[/tex]
[tex]\alpha = 0.05[/tex]
Calculate [tex]z_{\alpha/2}[/tex]
[tex]z_{\alpha/2} = z_{0.05/2}[/tex]
[tex]z_{\alpha/2} = z_{0.025}[/tex]
The z score is:
[tex]z_{\alpha/2} = z_{0.025} =1.96[/tex]
The endpoints of the confidence level is:
[tex](\bar x_1 - \bar x_2) \± z_{\alpha/2} * \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}[/tex]
[tex]2.0 \± 1.96 * \sqrt{\frac{2.1^2}{60}+\frac{3^2}{35}}[/tex]
[tex]2.0 \± 1.96* \sqrt{\frac{4.41}{60}+\frac{9}{35}}[/tex]
[tex]2.0 \± 1.96 * \sqrt{0.0735+0.2571}[/tex]
[tex]2.0 \± 1.96* \sqrt{0.3306}[/tex]
[tex]2.0 \± 1.1270[/tex]
Split
[tex](2.0 - 1.1270) \to (2.0 + 1.1270)[/tex]
[tex](0.8730) \to (2.1270)[/tex]
Hence, the 95% confidence interval is:
[tex]CI = (0.8730,2.1270)[/tex]
