standard error = standard deviation / square root of sample size.
use of an online calculator helps.
here's one for z-scores.
https://davidmlane.com/hyperstat/z_table.html
if you are using the population standard deviation, then use the z-score if the sample size is greater than 30, otherwise use the t-score.
here's a reference.
https://math.stackexchange.com/questions/1817980/how-to-know-when-to-use-t-value-or-z-value
your problem is using the standard deviation from the population, but the sample size is less than 30, so use of the t-score is indicated.
i'll do both so you can see the difference, but your official answer should be from the t-score unless the problem doesn't care whether you use the t-score or the z-score.
that you need to get from your professor.
using the z-score ********************************************
z = (x - m) / s
z is the z-score
x is the raw score
m is the raw mean
s is the standard error.
s = standard deviation from the population divided by square root of sample size = 1 / sqrt(16) = 1/4 = .25.
z-score for 5.5 million euros becomes:
z = (5.5 - 5) / .25 = 2
you will be looking for the area under the normal distribution curve to the right of a z-score of 2.
that area will be .0228 rounded to 4 decimal places. *****
the area to the right of the z-score is the probability that you will get a z-score greater than 2.
here is a graph of the results using the z-score.
z-score for 4.9 million euros becomes:
z = 4.9 - 5) / .25 = -.4
z-score for 5.1 million euros becomes:
z = (5.1 - 5) / .25 = .4
you will be looking for the area under the normal distribution curve between z = -.4 and z = .4
that area will be .31084 rounded to 5 decimal places.
same standard error is used.
here is a graph of the results using the z-score.
using the t-score **************************************
when you are using the t-score, you need to use the degrees of freedom.
for simple problem, such as this, the degrees of freedom are equal to the sample size minus of 1.
your degrees of freedom are 15.
your calculation of the t-score is the same, i.e......
t = (x - m) / s
x is the same.
m is the same.
s is the same.
what will be different is the area to under the normal distribution curve.
for area to the right of 5.5, the t-score will be 2.
the area to the right of that t-score with 15 degrees of freedom will be equal to .031972502 = .032 rounded to 3 decimal places, as in the graphical display below.
for area between t-score of -.4 and.4, the area in between will be equal to .3052050642 = .3052 rounded to 4 decimal places, as in the graphical display below.
the difference in the area between the z-score and the t-score are greater when the sample size is smaller and gets less as the sample size grows larger.
the z-score calculator can be found at https://davidmlane.com/hyperstat/z_table.html
the t-score calculator can be found at https://mathcracker.com/t-distribution-graph-generator#results
for my own calculations, i used the TI-84 plus calculator.
it provides results to a greater degree of accurac6y (more decimal places).