the confidence level is .98.
the sample mean is 7.4
the population standard deviation is .5
the sample size is 60.
if we let m = the population mean, then the z-score formula says that.
z = (x - m) / s
z is the z-score.
x is the mean of the sample
m is the mean of the population
s is the standard error of the distribution of sample means.
the formula for s would be:
s = standard deviation of the population divided by the square root of the sample size.
that would make s equal to .5/sqrt(60) = .0645497224 = .06455 rounded to 5 decimal places.
the formula becomes:
z = (7.4 - m) / .06455
with two tailed confidence level of 98%, the alpha on each end would be .01.
you would be looking for a z-score that has .99 area to the left of it or .01 area to the left of it.
that would make the z-score plus or minus 2.326 rounded to 3 decimal places.
when the z-score is plus 2.326, the formula becomes 2.326 = (7.4 - m) / .06455
multiply both sides of this equation by .06455 to get:
2.326 * .06455 = 7.4 - m
add m to both sides of this equation and subtract 2.326 * .06455 from both sides of this equation to get:
m = 7.4 - 2.326 * .06455 which makes m = 7.2498567
when the z-score is minus 2.326, the formula becomes -2.326 = (7.4 - m) / .06455.
multiply both sides of this equation by .06455 to get:
-2.326 * .06455 = 7.4 - m
add m to both sides of this equation and add -2.326 * .06455 to both sides of this equation to get:
m = 7.4 + 2.326 * .06455 which makes m = 7.5501433.
at 98% confidence level, the true population mean is assumed to be somewhere between 7.2498567 and 7.5501433.
this is what the 98% confidence interval looks like.
