Coefficient of Variation and z test vs t test

CFA Level I Quant
#1

  1. When calculating coefficient of variation (standard deviation/sample mean), is it the sample standard deviation or the population standard deviation we use? I have seen both being used and would like to know which one I should go with.

  2. From my understanding when n > 30 we can use the z test, i.e. we use the critical value for a normal distribution. Say if it was 5% two-tailed test with n > 30, we could use 1.96 critical value which approximates a normal distribution. However, there have been questions where n has been less than 30, e.g. 20 and instead of using the t critical value that reflects the degrees of freedom, i.e. in this case 2.086, my tutor still uses 1.96 in his calculations. Is my tutor wrong?

#2

My understanding, use like for like. Sample goes with sample, When n<30 and the sample variance is known, Z is approprate.

#3

One thing to consider is that the number 30 for a “sufficiently large” sample is somewhat “loose” (28 doesn’t necessitate a small sample really any more than 32 is large). As the sample size grows, the t-distribution becomes more and more like a z-distribution. I think your tutor may be using the z cutoffs to avoid looking up critical values of t more than he or she desires. I would say use the t-distribution in this case. Simply put and aside from appropriately accounting for the degrees of freedom, if you use the z cutoff of 1.96 when you should be using 2.086 (t cutoff), you will sometimes make the conclusion that you have sufficient evidence to reject the null hypothesis although you should not, after your degrees of freedom are appropriately accounted for; you will increase your probability of a Type I error without knowing it. Example:

I have a test statistic of 1.98

At the z-cut off, I have just barely shown significance. However, I should be using the t-cutoff, which at 2.086 would lead me to claim no statistical significance.

#4

So, in other words use the t-test by default unless the question specifically says that it is a normal distribution?

#5

I am not sure how the CFAI would have you do it (haven’t cracked my books yet), but in most cases using the t-distribution is a safer bet. As mentioned earlier, it conforms to the z distribution as n gets larger. Also, most statistical software packages present t-statistics in a majority of these cases. Personally, I would use a t-table unless it states otherwise or you’re only provided with a z-table.

Also, in using a z-table you are assuming the population standard deviation and the sample standard deviation are the same. Hence, you should use z when you know the distribution and the standard deviation.