Question 1201105
sample mean is 8.32 years
population standard deviation is 3.07 years.
z-score formula is z = (x-m)/s
z is the z-score
x is the upper and lower limit of the sample mean
x is the sample mean
s is the standard error


standard error = sample standard deviation / squre root of sample size = 3.07 / sqrt(40) = .4854096208.


t-score is indicated because the mean is taken from the sample, rather than from the population.
critical t-score with 39 degrees of freedom (sample size minus 1) is equal to 1.684875066.


t-score formula is t = (x-m)/s
t is the t-score
x is the estimated upper and lower limits of the population inferred from this sample based on the critical t-scores.
m is the smaple mean.
s is the standard error that is calculated from the sample standard deviation divided by the squre root of the sample size.


for the upper limit, the t-score formula becomes:
1.684875066 = (x - 8.32) / .4854096208
solve for x to get:
x = 1.684875066 * .4854096208 + 8.32 = 9.137854567.


for the lower limit, the t-score formula becomes:
-1.684875066 = (x - 8.32) / .4854096208
solve for x to get:
x = -1.684875066 * .4854096208 + 8.32 = 7.502145433.


your 90% confidence interval for how long current members of the national club of south valley have been members, calculated from the t-score, is 7.502145433 to 9.137854567 years.