Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of medical files is used to estimate the proportion p of all people who have blood type B.

(a) If you have no preliminary estimate for p, how many medical files should you include in a random sample in order to be 85% sure that the point estimate p̂ will be within a distance of 0.05 from p? (Round your answer up to the nearest whole number.)
medical files

(b) Answer part (a) if you use the preliminary estimate that about 13 out of 90 people have blood type B.
medical files

Answer 1

Solution :

Given that,

a) $\hat p$ = 1 - $\hat p$ = 0.5

margin of error = E = 0.05

At 85% confidence level

$\alpha$ = 1 - 85%

$\alpha$ =1 - 0.85 =0.15

$\alpha$/2 = 0.075

Z$\alpha$/2 = 1.44

sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (1.44 / 0.05)² * 0.5 * 0.5

= 207.36

sample size = n = 208

b) n = 90

x = 13

Point estimate = sample proportion = $\hat p$ = x / n = 13 / 90 = 0.14

1 - $\hat p$ = 1 - 0.14 = 0.86

sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (1.44 / 0.05)² * 0.14 * 0.86

= 99.86

sample size = n = 100