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2 answers:
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Answer:
i)
Sample size making use of the Census Bureau: 1,499 American adults.
Sample size without making use of the Census Bureau: 1,692 American adults
ii)
71
Step-by-step explanation:
i)
The sample size n in Simple Random Sampling is given by
where
<em>z = 1.645 is the critical value for a 90% confidence level </em><em>(*) </em>
<em>p= 0.669 is the population proportion given by the Census </em>
<em>e = 0.02 is the margin of error </em>
so
rounded up to the nearest integer.
(*)This is a point z such that the area under the Normal curve N(0,1) 1nside the interval [-z, z] equals 90% = 0.9
<em>It can be obtained with tables or in Excel or OpenOffice Calc with </em>
<em>NORMSINV(0.95) </em>
<em> </em>
If she ignores the Census estimate, the she has to take the largest sample possible that meets the requirements.
Let's show it is obtained when p = 0.5
As we said, the sample size n is
where
e = 0.02 is the error proportion
z = 1.645
hence
taking the <em>derivative</em> with respect to p, we get
n'(p)=6765.0625-2*6765.0625p
and
n'(p) = 0 when p=0.5
By taking the second derivative we see n''(p)<0, so p=0.5 is a maximum of n
<em>This means that if we set p=0.5, we get the maximum sample size for the confidence level required for the proportion error 0.02 </em>
Replacing p with 0.5 in the formula for the sample size we get
rounded to the nearest integer.
ii)
When we do not have a proportion but a variable whose approximate standard deviation s is known, then the sample size n in Simple Random Sampling is given by
where
<em>z = 2.241 is the critical value for a 95% confidence level </em><em>(*) </em>
<em>s = 7.5 is the estimated population standard deviation </em>
<em>e = 2 hours is the margin of error </em>
so
(*)This is a point z such that the area under the Normal curve N(0,1) inside the interval [-z, z] equals 95% = 0.95
<em>It</em> <em>can be obtained in Excel or OpenOffice Calc with </em>
<em>NORMSINV(0.9875) </em>