Question

A 90% confidence interval for a proportion is found to be (0.22, 0.28). What is

Answer 1

Answer:

The sample proportion is 0.253.

Step-by-step explanation:

We are given that a 90% confidence interval for a proportion is found to be (0.22, 0.28).

Firstly, as we know that the confidence interval for sample proportion is calculated as;

90% confidence interval = Sample proportion $\pm$ Margin of Error

Here, let $\hat p$ = sample proportion

Also, the level of significance = 1 - 0.90 = 0.10 or 10%

And, the critical value of z at 5% (two-sided) level of significance is 1.645.

So, 90% confidence interval =  $\hat p \pm 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$

           (0.22 , 0.28)    =   $\hat p \pm 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$

This means;  

          0.22  =  $\hat p - 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$  -------------- [Equation 1]

          0.28  =  $\hat p + 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$  -------------- [Equation 2]

From equation 1 and 2, we get;

           $0.22 + 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }= 0.28 - 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$

           $1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} } + 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }= 0.28 -0.22$

             $2 \times 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} } =0.06$

                 $\sqrt{\frac{\hat p(1-\hat p)}{n} } =\frac{0.06}{2 \times 1.645}$

                  $\sqrt{\frac{\hat p(1-\hat p)}{n} } =0.02$

Now, squaring both sides, we get;

                    ${\frac{\hat p(1-\hat p)}{n} } =0.0004$

                      $n ={\frac{\hat p(1-\hat p)}{0.0004} }$

Now, putting value of n in equation 1, we get;

                    0.22  =  $\hat p - 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{n} }$  

                    0.22  =  $\hat p - 1.645 \times \sqrt{\frac{\hat p(1-\hat p)}{\hat p(1-\hat p)}\times 0.0004 }$

                    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

Therefore, the sample proportion  $\hat p$ is 0.253.