Question 1118054
if you don't know the standard devition for the population, then you use the T-test rather than the Z-test.


your alpha = .01.


it's all on the right side of the normal distribution because your test is greater than rather than not equal.


your null hypothesis is that the mean is 59.5


your alternate hypothesis is that the mean is greater than 59.5.


your sample size is 16 with a mean of 68.1 and a standard deviation of 13.2.


you want to know what the critical value for this test is.


you need to look that up in the critical T-score Table.


your degrees of freedom are the sample size minus 1 = 15.


you are looking for a one-tail critical T-score with an alpha of .01.


the T-score Table i looked at is at <a href = "http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf" target = "_blank">http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>


here's a picture of what i found.


<img src = "http://theo.x10hosting.com/2018/060201.jpg" alt="$$$" >


i looked up a one tail probability of .01.
that was in the 8th column to the right of the first column which contained the title of one-tail.


i then looked down in the df column (degrees of freedom) until i found 15.


i then went to the 8th column to the right of that on that row and found a critical T-score of 2.602.


i then confirmed with my TI-84 Plus Calculator that the critical T-score was 2.602.


in that, the calculator gave me the answer to more decimal places (2.60248029).


my critical T-score is what i am testing against.


if my test T-score is greater than tht, then the results are significant and i would say that it is likely that the T-score of my test was not due to random variations in samples of size 16, but possibly something more significant, like a real difference dues to something other than chance.


the critical alpha of = .01 means that i can be fairly confident that, if i did this test an infinite number of times., that the test T-score would be within the confidence limits 99% of the time, meaning that only 1 out of 100 samples would give me the results i saw.


to do the test, i need to calculate the standard error.


the standard error is the standard deviation of the distribution of sample means.


the standard error is affected by the sample size.


the larger the sample size, the smaller the standard error.


if i can find a reference to explain this to you, i'll include it down below.


for now, just accept that it is so.


the formula for sample error is s = sqrt(sd/n)


s is the standard error.
sd is the standard deviation of the population or the sample.
n is the sample size.


if you know the standard deviation of the population, you use that.


if not, you use the standard deviation of the sample itself.


using this formula, you get s = sqrt(13.2/16) = .9082951062.


i also need to calculate the T-score for the test, using that standard error.


the T-score formula is T(15) = (x-m)/s.


x is the mean of the sample.
m is the mean of the population, or the assumed mean of your null hypothesis.
s is the sample size.


it's the same as the Z-score formula, except you are dealing with a T-score and not a Z-score, and you have to take into consideration the degrees of freedom.


using the information i know have, i get:


T(15) = (68.1 - 59.5) / .9082951062 = 9.46828838.


my critical T-score is 2.602 for samples of size 16.


this is much greater than that, so i can safely assume that the difference is not due to random variation in samples of size 16, but a real difference due to something other than chance.


here's a reference on sample error.


check it out and then go to the next section on the central limit theorem.


<a href = "http://davidmlane.com/hyperstat/A103397.html" target = "_blank">http://davidmlane.com/hyperstat/A103397.html</a>


if you go a bit further, there is even a simulation that you might be able to perform that demonstrates the central limit theorem in action.


the rules for when you would use a T-score rather than a Z-score are discussed in the following reference.


<a href = "http://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/" target = "_blank">http://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/</a>