Question

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

The National Council of Small Businesses is interested in the proportion of small businesses that declared Chapter 11 bankruptcy last year. Since there are so many small businesses, the National Council intends to estimate the proportion from a random sample. Let p be the proportion of small businesses that declared Chapter 11 bankruptcy last year.

(a) If no preliminary sample is taken to estimate p, how large a sample is necessary to be 95% sure that a point estimate p̂ will be within a distance of 0.08 from p? (Round your answer up to the nearest whole number.)
small businesses

(b) In a preliminary random sample of 30 small businesses, it was found that ten had declared Chapter 11 bankruptcy. How many more small businesses should be included in the sample to be 95% sure that a point estimate p̂ will be within a distance of 0.080 from p? (Round your answer up to the nearest whole number.)
more small businesses

Answer 1

solution :

Given that,

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

margin of error = E = 0.08

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = 1.96

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

= (1.96/ 0.08)² * 0.5 * 0.5

= 150.06

sample size = n = 151

n = 30

x =10

Point estimate = sample proportion = $\hat p$ = x / n = 10 / 30 = 0.33

1 - $\hat p$ = 1-0.33 = 0.67

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

= (1.96/ 0.08)² * 0.33 * 0.67

= 132.71

sample size = n = 133