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

The minimum sample size is
`n =135`

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 = ( upper \ limit + lower \ limit )/(2)`

`\r p = ( 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
`(\alpha )/(2)` from the normal distribution table

The value is

`Z_{(\alpha )/(2) } = 2.33`

Generally the minimum sample size is evaluated as

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

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

`n =135`