Question

Express the confidence interval 0.555 less than 0.777 in the form Modifying above p with caret plus or minus Upper E.

Answer 1

Complete Question

Express the confidence interval 0.555 less than p less than  0.777 in the form Modifying above p with caret plus or minus Upper E.

Answer:

The modified representation is $\r p \pm E = 0.666 \pm 0.111$

Step-by-step explanation:

From the question we are told that

    The  confidence interval interval is  $0.555 < p < 0.777$

Now looking at the values that make up  the up confidence interval we see that this is a symmetric  confidence interval(This because the interval covers 95%  of the area under the normal curve which mean that the probability of a value falling outside the interval is 0.05 which is divided into two , the first half on the left -tail and the second half on the right tail  as shown on the figure in the first uploaded image(reference - Yale University ) ) which means

     Now  since the confidence interval is  symmetric , we can obtain the sample proportion as follows

             $\r p = \frac{0.555 + 0.777}{2}$

            $\r p =0.666$

Generally the margin of error is mathematically represented as

            $E = \frac{1}{2} * K$

Where  K is the length of the confidence interval which iis mathematically represented as

           $K = 0.777 -0.555$

         $K = 0.222$

Hence  

          $ME = \frac{1}{2} * 0.222$

           $ME = 0.111$

So the confidence interval can now be represented as

        $\r p \pm E = 0.666 \pm 0.111$