Question 1192947
population mean is assumed to be 24.
sample mean is 21.5
sample standard deviation is 1.9.
sample size is 29.


standard error is 1.9 / sqrt(29) = .352821 rounded to 6 decimal places.


z = (x - m) / s
z is the z-score
x is the sample mean
m is the population mean
s is the standard error.


the formula becomes z = (21.5 - 24) / .352821 = -7.08574603.


the area to the left of that z-score is equal to 0.


what this says is that the probability you would get a z-score less than -7.08..... is so small as it is effectively equal to 0.


here's a reference on symbols.


<a href = "https://www.dummies.com/article/academics-the-arts/math/statistics/seeing-what-statistical-symbols-stand-for-142633/" target = "_blank">https://www.dummies.com/article/academics-the-arts/math/statistics/seeing-what-statistical-symbols-stand-for-142633/</a>


note that, because you are using the sample standard deviation rather than the population standard deviation, you would probably be using the t-stoce rather than the z-score.


the t-score itself is calculated in the same way as the z-score.


the difference is in the area under the normal distribution curve to the left or right of the t-score as opposed to the area to the left or right of the z-score.


t-scores use degrees of freedom whereas z-score don't.


the degrees of freedom are usually 1 less than the sample size.


if the degrees of freedom are small, there is a larger difference between the area to the left or right of a t-score compared to the area to the left or right of a z-score.


some examples:


z-score = -1
area to the left = .158655


t-score with 5 degrees of freedom = -1
area to the left = .181609


t-score with 50 degrees of freedom = -1
area to the left = .161063


t-score with 1000 degrees of freedom = -1
area to the left = .158776


as the degrees of freedom get greater, the area to the left of the t-score approaches the area to the left of the z-score.


the difference is most pronounced when the degrees of freedom are small.


here's a reference that shows you the difference between a t distribution and a z distribution.


<a href = "https://www.jmp.com/en_us/statistics-knowledge-portal/t-test/t-distribution.html#:~:text=What's%20the%20key%20difference%20between,on%20the%20sample%20standard%20deviation." target = "_blank">https://www.jmp.com/en_us/statistics-knowledge-portal/t-test/t-distribution.html#:~:text=What's%20the%20key%20difference%20between,on%20the%20sample%20standard%20deviation.</a>