Question 1198931: A survey is being planned to determine the mean amount of time children under 6 spend watching TV.
A pilot survey indicated the mean time was 22 hours per week with a standard deviation of 3 hours.
The mean viewing time is to be estimated within 0.5 hours. How many children must be studied to
obtain a 95% confidence interval?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! population mean = 22
population standard deviation = 3
standard error = standard deviation / square root of sample size.
let se = standard error and sd = standard deviation and n = sample size.
you get se = sd / sqrt(n)
use the z-score formula to solve for this.
z is the t-score
x is the sample mean
m is the population mean
s is the standard error
the margin of error is equal to plus or minus (x - m).
when the margin of error is .5, then (x - m) = .5
critical z-score at two tail 95% confidence interval is plus or minus 1.644853626.
on the high side, z-score formula becomes 1.644853626 = .5 / (sd / sqrt(n)) which becomes:
1.644853626 = .5 / (3/sqrt(n))
multiply both sides by (3/sqrt(n)) to get:
3/sqrt(n) * 1.644853626 = .5
multiply both sides by sqrt(n) and divide both sides by .5 to get:
3 * 1.644853626 / .5 = sqrt(n)
solve for sqrt(n) to get:
sqrt(n) = 9.869121756
solve for n to get:
n = 97.39956423.
since n has to be an integer, then round up to get n = 98.
that should be your sample size that will make the margin of error less than .5
it will not be equal to .5 because, for that to happen, the sample size would have to be 97.39956423.
standard error becomes 3/sqrt(8).
go back to your original equation to see if this works.
in your original problem, on the high side, you get:
1.644853626. = (x - 22) / (3/sqrt(98))
solve for x to get:
x = 1.644853626 * 3/sqrt(98) + 22 = 22.49846592
in your original problem, on the low side, you get:
-1.644853626 = (x - 22) / (3/sqrt(98))
solve for x to get:
x = -1.644853626 * 3/sqrt(98) + 22 = 21.50153408
your 95% confidence interval is between 21.50153408 and 22.49846592
your margin of error is less than .5
if you could have used a sample size of n = 97.39956423, then the margin of error would have been exactly plus or minus .5
for example, on the high side, you get:
1.644853626 = (x - 22) / (3/sqrt(97.39956423).
solve for x to get:
x = 1.644853626 * 3/sqrt(97.39956423) + 22 = 22.5
on the low side you would get:
x = -1.644853626 * 3/sqrt(97.39956423) + 22 = 21.5
your margin of error is eaxactly .5
but !!!!!, .....
your sample size has to be integer, so the smallest your sample size can be is 98 which will get you a margin of error less than .5
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