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 0.11 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 fourteen 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 0.110 from p? (Round your answer up to the
nearest whole number.)
more small businesses
Solution :
Given that,
a) = 1 - = 0.5
margin of error = E = 0.11
At 99% confidence level
= 1 - 99%
=1 - 0.99 =0.01
/2
= 0.005
Z/2
= 2.576
sample size = n = (Z / 2 / E )2 * * (1 - )
= (2.576 / 0.11)2 * 0.5 * 0.5
= 137.10
sample size = n = 138
b) Given that,
n = 30
x = 14
Point estimate = sample proportion = = x / n = 14 / 30 = 0.47
1 - = 1 - 0.47 = 0.53
sample size = n = (Z / 2 / E )2 * * (1 - )
= (2.576 / 0.11)2 * 0.47 * 0.53
= 136.60
sample size = n = 137
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