Question 1200882: A section manager wishes to estimate the mean number of seconds required by a worker to do a particular task. He observed the worker on 144 randomly selected occasions. The average number of seconds required in the 144 observations was 100 seconds and the standard deviation was 10 seconds. What size of sample (i.e. how many observations) would be necessary to estimate the true mean within an error of 0.5 second with a 95 percent confidence coefficient? (Use the standard deviation of the sample as the best available estimate of the standard deviation of the population).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sample size is 144.
sample mean is 100 seconds.
sample standard deviation is 10 seconds.
you want a margin of error of plus or minus .5 seconds.
you're looking for 95% confidence interval.
you're using the sample standard deviation as the best available estimate of the standard deviation of the population.
if you assume the sample standard deviation is the same as the population standard deviation, then you can use the z-score.
you would find the critical z-score for 95% confidence interval.
that would be equal to plus or minus z = 1.96.
you want the margin of error to be equal to plus or minus .5.
that means that (x-m) would have to be equal to plus or minus .5.
the z-score formula is z = (x-m) / s
z is the z-scre
x is the sample mean
m is the population mean
plus or minus (x-m) is the margin of error
s is the standard error.
standard error is equal to standard deviation divided by square root of sample size.
on the high side, your z-score formula becomes 1.96 = .5 / s.
solve for s to get s = .5 / 1.96.
s = 10/sqrt(n)
n is the sample size]
formula becomes 10/sqrt(n) = .5/1.96
solve for sqrt(n) to get sqrt(n) = 10 * 1.96 / .5 = 39.2
solve for n to get n = 39.2^2 = 1536.64.
with that sample size, your standard error becomes 10/sqrt(1536.64) = .2551020408.
with a mean of 100 seconds, the z-score formula becomes:
1.96 = (x-100)/.2551020408.
solve for x to get:
x = .2551020408 * 1.96 + 100 = 100.5
with a z-score of -1.96, you get x = -1.96 * .2551020408 + 100 = 99.5.
since your sample size needs to be an integer, you would choose the next higher integer for the sample size.
the next higher integer for the sample size would be 1537.
sqrt(1537) = 39.20459157.
the standard error would become 10/that = .2550721637.
with that standare error, the high side would be 1.96 = (x-100)/.2551020408.
solve for x to bet 100.4999414.
that's something less than .5.
you will get a correspondingly margin of error on the low side as well (margin of error < .5).
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