Question 1206548: he mean number of travel days per year for salespeople employed by three hardware distributors needs to be estimated with a 0.90 degree of confidence. For a small pilot study, the mean was 150 days and the standard deviation was 14 days. If the population mean is estimated within two days, how many salespeople should be sampled. a. 133 b. 452 c. 511 d. 2,100
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! confidence interval is 90%.
mean = 150 days
standard deviation = 14 days.
standard error = standard deviation / sqrt(sample size) = 14/sqrt(n)
90% confidence interval requires z-score equal to plus or minus 1.6449.
z-score formula is z = (x-m)/s
z is the critical z-score
x is the critical raw score
m is the raw mean
s is the standard error.
with critical z-score of plus or minus 1.6445 and a margin of error of 2 and a standard error of 14 / sqrt(n), the formula becomes:
1.6445 = 2 / (14 / sqrt(n))
this is equivalent to 1.6445 = 2 * sqrt(n) / 14
multiply both sides by 14 and divide both sides by 2 to get 1.6445 * 14 / 2 = sqrt(n) = 11.5115.
with that as the standard error, you should get a margin of error 2.
let's see.
on the high side of the confidence interval, z = (x - m) / s becomes 1.6445 = (x - 150) / (14 / 11.5115).
solve for x to get x = 1.6445 * 14 / 11.5115 + 150 = 152.
margin of error is 152 minus 150 = 2.
so far so good.
on the low side of the confidence interval, z = (x - m) / s becomes -1.6445 = (x - 150) / (14 / 11.5115).
solve for x to get x = -1.6445 * 14 / 11.5115 + 150 = 148.
margin of eror is 148 - 152 = -2.
that's good.
your margin of error is 2.
that means your sample mean is between 148 and 152 at 90% confidence interval.
one more thing to do.
sample size is sqrt(11.5115) ^2 rounded to the next highest integer.
that will get you 132.5 rounded to the next highest integer = 133.
it has to be rounded to the next highest integer because fractional sample sizes are not allowed.
the smallest your sample size can be is 133.
that will allow you to get a margin of error slightly less than 2.
after your rounded the sample size to the next highest integer, you would need to re-calculate the standard error.
standard error = 14 / sqrt(133) = 1.213954 rounded to 6 decimal places.
with that standard error, your margin of error formula becomes 1.6445 = (x - m) / 1.213954 = 1.996347.
that's slightly less than a margin of error of 2.
it's the smallest sample size that will allow you to get a margin of error less than or equal to 2.
133 looks like selection a.
that's your solution.
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