Respuesta :
The value of sample proportion is found to be 0.253.
What is the sample proportion?
The sample proportion [tex]\hat{p}[/tex] indicates the percentage of people in a sample who exhibit a particular quality or attribute.
Divide the number of individuals (or items) who possess the desired attribute by the overall sample size to obtain the sample proportion.
Now, according to the question;
The 90% confidence interval for a proportion is found to be (0.22,0.28).
The confidence interval for the sample is given as-
Confidence interval = Sample proportion ± Margin of Error
Let [tex]\hat{p}[/tex] be the sample proportion.
The level of significance = 1 - 0.90
= 0.10 (10%)
From z-value table, at a significance level of 5% (two-sided), z has a critical value of 1.645.
90% confidence interval = [tex]\hat{p} \pm 1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex]
Thus, 0.22 = [tex]\hat{p} - 1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex], and
0.28 = [tex]\hat{p} + 1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex]
Covert both equation in the form of [tex]\hat{p}[/tex] and equating to each other.
[tex]\begin{aligned}&0.22+1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=0.28-1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\\&1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}+1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=0.28-0.22\\&2 \times 1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=0.06\\&\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\frac{0.06}{2 \times 1.645}\\&\sqrt{\frac{\hat{\nu}(1-\hat{p})}{n}}=0.02\end{aligned}[/tex]
Take square root,
[tex]\begin{gathered}\frac{\hat{p}(1-\bar{p})}{n}=0.0004 \\n=\frac{\hat{p}(1-\bar{p})}{0.0004}\end{gathered}[/tex]
Substituting the value of 'n' in the receptive equation to get the value of [tex]\begin{aligned}0.22 &=\hat{p}-1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\0.22 &=\hat{p}-1.645 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{p(1-\hat{p})}} \times 0.0004 \\0.22 &=\hat{p}-1.645 \times \sqrt{0.0004} \\0.22 &=\hat{p}-(1.645 \times 0.02) \\0.22 &=\hat{p}-0.033 \\\hat{p}=& 0.22+0.033=0.253\\\end{aligned}[/tex]
Therefore, the value of sample proportion is 0.253.
To know more about the sample proportion, here
https://brainly.com/question/14089562
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