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Evaluate [tex]\( E = z^\ \textless \ em\ \textgreater \ \sqrt{\frac{\hat{p}_1 \cdot \hat{q}_1}{n_1} + \frac{\hat{p}_2 \cdot \hat{q}_2}{n_2}} \)[/tex] for [tex]\( z^\ \textless \ /em\ \textgreater \ = 1.645 \)[/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], [tex]\( \hat{p}_2 = 0.0022 \)[/tex], and [tex]\( \hat{q}_2 = 0.9978 \)[/tex].

[tex]\[ E = \][/tex] [tex]\(\square\)[/tex] (Round to four decimal places as needed.)



Answer :

GinnyAnswer
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]

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