that's because the mean of a sample of size greater than 1 element requires the use of the standard error, which is the standard deviation of the population divided by the square root of the sample size.
if you are talking about a sample element, then you are talking about the value of that sample element, and not the mean of that sample.
believe it or not, if the sample size is 1 element, then the mean of the sample is the value of the sample element itself and the standard error is equal to the standard deviation of the population.
i will assume you are talking about individual samples rather than the mean of a sample, since you did not specify the sample size (number of elements in the sample).
with that understanding, this is what will happen.
the population mean is 116000 and the population standard deviation is 36000.
the z-score formula is z = (x-m) / s
z is the z-score.
x is the value of the sample
m is the mean of the population
s is the standard deviation of the population.
answers to your quewtions will be shown below:
c. What is the likelihood of selecting a sample with a mean of at least $122,000? (Round your z-value to 2 decimal places and final answer to 4 decimal places.)
z-score becgomes z = (122000 - 116000) / 36000 = .17
area to the right of a z-score of .17 = .4325.
that's the probability that you will get a sample whose value is greater than 122000.
d. What is the likelihood of selecting a sample with a mean of more than $106,000? (Round your z-value to 2 decimal places and final answer to 4 decimal places.)
z = (x - m) / s becomes z = (106000 - 116000) / 36000 = -.27
area to the right of a z-score of -.27 = .6064.
that's the probability that you will get a sample whose value is greater than 106000.
e. Find the likelihood of selecting a sample with a mean of more than $106,000 but less than $122,000. (Round your z-value to 2 decimal places and final answer to 4 decimal places.)
z-score for 106000 = -.27
z-score for 122000 = .17
area to the left of z-score of -.27 = .3936.
area to the left of z-score of .17 = .5675.
area in between = .5675 minus .3936 = .1739.
that's the probability that you will get a sample shose value is between 106000 and 122000.
this is wht it looks like on a graph.
let me know if you have any questions.
theo