Normal Distribution Hypothesis Tests

When to do a Normal Hypothesis Test

There are two types of hypothesis tests you need to know about: binomial distribution hypothesis tests and normal distribution hypothesis tests. In binomial hypothesis tests, you are testing the probability parameter p. In normal hypothesis tests, you are testing the mean parameter \mu. This gives us a key difference that we can use to determine what test to do and when.

Since normal hypothesis tests test a mean parameter, words like mean, average and overall are all clues that you should use a normal hypothesis test. The situation and the context should also help – if you would model it with a normal distribution, you want to do a normal hypothesis test.

How to do a Normal Hypothesis Test

1. Define the parameter in the context of the question – for a normal hypothesis test the parameter is \mu which is always the mean of something.
2. Write down the null hypothesis and the alternate hypothesis.
3. Define the test statistic X in the context of the question.
4. Write down the distribution of X under the null hypothesis.
5. State the significance level \alpha – even though you are likely given it in the question, not stating it risks losing a mark.
6. Test for significance or find the critical region.
7. Write a concluding sentence, linking the acceptance or rejection of H_{0} to the context.

Handling Multiple Observations

If we are given multiple observations on which to base our hypothesis test, we can produce a more accurate result and have a larger critical region.

If we have a normally distributed variable X\sim N(\mu,\sigma^{2}), the average of n observations of X has the distribution \bar{X}\sim N\left(\mu,\dfrac{\sigma^{2}}{n}\right).

This means that if we are given multiple observations we can do the hypothesis test with the average of the observations rather than a single observation. Indeed, this is much better than using a single observation because it has a larger critical region.