Which Distribution?

Quite often in intro statistics courses, you are challenged with a problem like this:

#1: Some homeowners claim that the mean speed of cars traveling on their street is greater than the speed limit of 35 miles per hour. A random sample of 100 cars has a mean speed of 36 miles per hour. Assume the population standard deviation is 4 miles per hour. Is there enough evidence to support the claim at α = 0.05?

So, this is a problem about a mean and to answer it we must use the right distribution – the normal z distribution or the t-distribution. That would mean we use the “z-test” or the “t-test.”

Here are two more example problems:

#2: A politician claims that the mean expenditure per student in public schools in their state is more than $12,000. You want to test this claim. You randomly select 16 school districts in the state and find the mean spent is $12,601 with a standard deviation of $491. At a = 0.01, can you support the politician’s claim?

#3: A study says the mean time to recoup the cost of bariatric surgery is 3 years.  You randomly select 25 bariatric surgery patients and find that the mean time to recoup the cost of their surgeries is 3.3 years.  Assume the population standard deviation is 0.5 year and the population is normally distributed.  Is there enough evidence to doubt the study’s claim if alpha is 0.10?

There are subtle differences in the three questions that you need to “see” if you are to get the correct answer. For most textbooks I have seen, this decision tree provides a good path to follow  to help you pick up on the key differences by asking three questions in sequence:

z vs t decision tree


Ask these questions in this order:

  1. Is the sample size at least 30?
    Many authors call the z-test the “large sample” test and the t-test the “small sample test.” According to the Central Limit Theorem, if the sample size is at least 30, the sampling distribution is normal even if the target population is not normal. Generally, if you can use the z-test, you should. So, if the sample size is at least 30, use the z-test.
  2. If the sample size is less than 30, then we need to consider what we know about the target population. If the question tells you to assume it is normally distributed, then you go to question 3. If you are not told the population is normal, then you do not run a z or a t-test. In more advanced courses, we would move to what is called a “non-parametric” test which is suitable for non-normal populations with small samples.
  3. So, if we know the population is normally distributed, we can ask the 3rd question: is the population standard deviation, sigma, known? If you know sigma, use the z-test. If not, use the t-test.


So, let’s review those three problems. 

In problem 1, we have a sample size of 100, which is much larger than 30.So we use the z-test.

In problem 2, we have a small sample size of 16. Thus we need to check for the population’s normality. Here we are not told it is normal, so we cannot do the z or the t-test. 

Note, in the “real world” I would plot a histogram of the sample means to see if it is close to normal, with no severe skew either way. If it is, then I would use the z-test. Check with your instructor to be sure of their preference.

In the 3rd problem,  we have a “small” sample of 25, but we are told to assume the population is normally distributed. Hence, we can use the z-test if we know sigma. Unfortunately, we do not know sigma, so we use the t-distribution.

Hope this helps!

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