Question 1201088
population standard deviation is 10.2
confidence interval is 95%
margin of error is smaller than or equal to 1.7
critical z-score is plus or minus 1.9599963986.
x-score formula is z = (x - m) / s
z is the z-score
x is the sample mean
m is the population mean
(x - m) is the margin of error
s is the standard error
the high side critical z-score is 1.9599963986.
z-score formula becomes:
1.9599963986 = (x - m) / s
s is the standard error.
standard error is standard deviation divided by square root of sample size.
formula become:
1.9599963986 = 1.7 / s
solve for s to gets = 1.7 / 1.9599963986 = .8673628761.
formula for standard error is s = standard deviation / sqrt(sample size)
you get:
.8673628761 = 10.2 / sqrt(sample size)
solve for sqrt(sample size) to get:
sqrt(sample size) = 10.2 / .8673628761 = 11.75978392.
solve for sample size to get:
sample size = 11.75978392 squared = 138.2925177.
with a sample size of 138.2925177, you should get a margin of error equal to 1.7.
round sample size to the next highest integer to get 139.
with a sample size of 139, you should get a margin of error less than 1.7.
a sample size of 139 gets you a standard error of 10.2 / sqrt(139) = .8651527083.
that standard error gets you a margin of error equal to 1.695668151.
this is very close to, but less than, a margin of error of 1.7.