Question

Find the minimum sample size needed to estimate the percentage of Democrats who have a sibling. Use a 0.1 margin of error, use a confidence level of 98%, and use the results from a prior Harris poll that gave a confidence interval of (0.44, 0.51) for the proportion of Democrats who have a sibling.

Answer 1

Answer:

The minimum sample size is  $n =135$

Step-by-step explanation:

From the question we are told that

   The confidence interval is $( lower \ limit = \ 0.44,\ \ \ upper \ limit = \ 0.51)$

    The margin of error is  $E = 0.1$

   

Generally the sample  proportion can be mathematically evaluated as

     $\r p = \frac{ upper \ limit + lower \ limit }{2}$

    $\r p = \frac{ 0.51 + 0.44}{2}$

    $\r p = 0.475$

Given that the confidence level is  98% then the level of significance can be mathematically evaluated as

         $\alpha = 100 - 98$

        $\alpha = 2\%$

        $\alpha =0.02$

Next we obtain the critical value of  $\frac{\alpha }{2}$ from the normal distribution table  

   The value is

        $Z_{\frac{\alpha }{2} } = 2.33$

Generally the minimum sample size is evaluated as

      $n =[ \frac { Z_{\frac{\alpha }{2} }}{E} ]^2 * \r p (1- \r p )$

     $n =[ \frac { 2.33}{0.1} ]^2 * 0.475(1- 0.475 )$

     $n =135$