To evaluate the expression [tex]\( E = z^ \sqrt{\frac{\hat{p}_1 \cdot \hat{q}_1}{n_1}+\frac{\hat{p}_2 \cdot \hat{q}_2}{n_2}} \)[/tex] using the given values:
- [tex]\( z^ = 1.645 \)[/tex]
- [tex]\( n_1 = 2799 \)[/tex]
- [tex]\( n_2 = 7799 \)[/tex]
- [tex]\( \hat{p}_1 = 0.0125 \)[/tex]
- [tex]\( \hat{q}_1 = 0.9875 \)[/tex] (since [tex]\( \hat{q}_1 = 1 - \hat{p}_1 \)[/tex])
- [tex]\( \hat{p}_2 = 0.0022 \)[/tex]
- [tex]\( \hat{q}_2 = 0.9978 \)[/tex] (since [tex]\( \hat{q}_2 = 1 - \hat{p}_2 \)[/tex])
1. Calculate the two components inside the square root:
[tex]\[
\frac{\hat{p}_1 \cdot \hat{q}_1}{n_1} = \frac{0.0125 \cdot 0.9875}{2799}
\][/tex]
and
[tex]\[
\frac{\hat{p}_2 \cdot \hat{q}_2}{n_2} = \frac{0.0022 \cdot 0.9978}{7799}
\][/tex]
2. Add these two components together:
[tex]\[
\frac{0.0125 \cdot 0.9875}{2799} + \frac{0.0022 \cdot 0.9978}{7799} \approx 4.6915 \times 10^{-6}
\][/tex]
3. Take the square root of this sum:
[tex]\[
\sqrt{4.6915 \times 10^{-6}} \approx 0.0021577
\][/tex]
4. Multiply by [tex]\( z^* \)[/tex]:
[tex]\[
E = 1.645 \times 0.0021577 \approx 0.0035631
\][/tex]
5. Round the result to four decimal places:
[tex]\[
E \approx 0.0036
\][/tex]
So, the value of [tex]\( E \)[/tex], rounded to four decimal places, is:
[tex]\[
E = 0.0036
\][/tex]
