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I have a data set about 5,699 PhD students. The first column called "Year" is how many years it took for candidate to graduate with a Ph.D (1, 2,...14), the second column "Uni" is which university the student received the PhD, and the third column "Res" is residency of subject (permanent or temporary)

> head(mydata)
   Year     Uni       Res
1    1 Berkeley Permanent
2    1 Berkeley Permanent
3    1 Berkeley Permanent
4    1 Berkeley Permanent
5    1 Berkeley Permanent
6    1 Berkeley Permanent

I wish to see if there is a significant difference in the number of years it took for PhD students to graduate by residency. I'm assuming I must perform a two sample t-test or a two sample z-test. I know that one performs a z-test if the standard deviation of the population is known, and a t-test if it is not known. However, I have no information on whether these 5,699 students form a population or if they are samples from a larger population. Since I am not sure, should I perform the two sample t-test?

One of the assumptions of the two sample t-test and the two sample z-test is that the data must be normally distributed. Does this mean I have check if the number of years to graduate is normal for each group (permanent, temporary) or do I combine the data and check to see if the years to graduation is normal? What kind of tests do I use to check for normality in this case? Are there other assumptions I should be aware about?

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  • $\begingroup$ Whether a group forms a sample or a population depends crucially on the intended scope of your inferences: are you aiming to describe these 5699 people or will you attempt to derive conclusions about a larger group of people from these data? BTW, neither test assumes data are normally distributed. You would benefit from reading some of our high-voted threads about these tests. $\endgroup$
    – whuber ♦
    Commented Oct 10, 2019 at 12:34
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    $\begingroup$ I'm wondering about the range and discreteness of the dependent variable (Year). If there are only a few values this variable can take, you might need an analysis appropriate for ordinal dependent variables... or maybe one appropriate for count variables. $\endgroup$
    – Sal Mangiafico
    Commented Oct 10, 2019 at 12:51
  • $\begingroup$ @whuber: Doesn't t-test assume normality? The sample statistic in t-test follows t-distribution under the assumption of normality isn't it? Asymptotically it will follow normal but to get the sampling distribution isn't normality a required assumption? $\endgroup$
    – Dayne
    Commented Oct 11, 2019 at 6:55
  • $\begingroup$ @Dayne The t-test assumes the sampling distribution of the statistic is sufficiently close to a Student t to permit the use of the Student t CDF in converting the t-statistic to a p-value for t-statistics close to the critical value. These italicized conditions tend to hold for many underlying non-Normal distributions and typical critical values (like 5% and 1%), especially those distributions that aren't very skewed. $\endgroup$
    – whuber ♦
    Commented Oct 11, 2019 at 14:01
  • $\begingroup$ @whuber can you please cite some good source for this? Also, you say t-test assumes this. I think this may be a result found later. The original does assume normality, afaik. $\endgroup$
    – Dayne
    Commented Oct 11, 2019 at 16:26

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