Respuesta :
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
[tex] \bf n=\frac{z^2p(1-p)}{e^2}[/tex]
where
z = 1.645 is the critical value for a 90% confidence level (*)
p= 0.669 is the population proportion given by the Census
e = 0.02 is the margin of error
so
[tex] \bf n=\frac{(1.645)^2*0.669*0.331}{0.02^2}=1,498.05\approx 1,499[/tex]
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
It can be obtained with tables or in Excel or OpenOffice Calc with
NORMSINV(0.95)
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
[tex] \bf n=\frac{z^2p(1-p)}{e^2}[/tex]
where
e = 0.02 is the error proportion
z = 1.645
hence
[tex] \bf n=\frac{(1.645)^2p(1-p)}{(0.02)^2}=6765.0625p(1-p)=6765.0625p-6765.0625p^2[/tex]
taking the derivative 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
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
Replacing p with 0.5 in the formula for the sample size we get
[tex] \bf n=6765.0625*0.5-6765.0625(0.5)^2=1691.27\approx 1,692[/tex]
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
[tex] \bf n=\frac{z^2s^2}{e^2}[/tex]
where
z = 2.241 is the critical value for a 95% confidence level (*)
s = 7.5 is the estimated population standard deviation
e = 2 hours is the margin of error
so
[tex] \bf n=\frac{z^2s^2}{e^2}=\frac{(2.241)^2(7.5)^2}{(2)^2}=70.62\approx 71[/tex]
(*)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
It can be obtained in Excel or OpenOffice Calc with
NORMSINV(0.9875)
Answer:
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
Step-by-step explanation:
