Sure, let's evaluate the given expression step-by-step:
We start with the given formula:
[tex]\[ E = z^ \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \][/tex]
Given values:
- [tex]\(z^ = 1.10\)[/tex]
- [tex]\(\sigma_1 = 0.77\)[/tex]
- [tex]\(\sigma_2 = 0.71\)[/tex]
- [tex]\(n_1 = 15\)[/tex]
- [tex]\(n_2 = 60\)[/tex]
Step 1: Calculate [tex]\(\frac{\sigma_1^2}{n_1}\)[/tex]
[tex]\[
\sigma_1^2 = 0.77^2 = 0.5929
\][/tex]
[tex]\[
\frac{\sigma_1^2}{n_1} = \frac{0.5929}{15} \approx 0.039527
\][/tex]
Step 2: Calculate [tex]\(\frac{\sigma_2^2}{n_2}\)[/tex]
[tex]\[
\sigma_2^2 = 0.71^2 = 0.5041
\][/tex]
[tex]\[
\frac{\sigma_2^2}{n_2} = \frac{0.5041}{60} \approx 0.0084016667
\][/tex]
Step 3: Add the two variances to get the total variance term
[tex]\[
\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} \approx 0.039527 + 0.0084016667 = 0.04792833333333334
\][/tex]
Step 4: Calculate the square root of the variance term
[tex]\[
\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{0.04792833333333334} \approx 0.21892540586540735
\][/tex]
Step 5: Multiply by [tex]\(z^*\)[/tex]
[tex]\[
E = 1.10 \times 0.21892540586540735 \approx 0.2408179464519481
\][/tex]
Step 6: Round the result to two decimal places
[tex]\[
E \approx 0.24
\][/tex]
Therefore, the value of [tex]\(E\)[/tex], rounded to two decimal places, is:
[tex]\[ E = 0.24 \][/tex]
