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 99% sure that a point estimate p̂ will be within a distance of 0.09from 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 99% sure that a point estimate p̂ will be within a distance of 0.090 from p? (Round your answer up to the nearest whole number.)
more small businesses

Answer 1

a)

The following information is provided,
Significance Level, α = 0.01, Margin of Error, E = 0.09

The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.01 is 2.576.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.5*(1 - 0.5)*(2.576/0.09)^2
n = 204.8079

Therefore, the sample size needed to satisfy the condition n >= 204.8079 and it must be an integer number, we conclude that the minimum required sample size is n = 205
Ans : Sample size, n = 205

b)

The following information is provided,
Significance Level, α = 0.01, Margin of Error, E = 0.09

The provided estimate of proportion p is, p = 0.3333
The critical value for significance level, α = 0.01 is 2.576.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.3333*(1 - 0.3333)*(2.576/0.09)^2
n = 182.0424

Therefore, the sample size needed to satisfy the condition n >= 182.0424 and it must be an integer number, we conclude that the minimum required sample size is n = 183
Ans : Sample size, n = 183