Hypothesis Testing: Normal Distribution

Here’s a reminder of the key points you should know about testing a population mean.

Population means and sample means are often different. 

But the bigger the sample, the closer the sample mean is likely to be to the population mean.

If a population is modelled using a normal distribution…

It’s possible to model the probabilities of observing different sample means like this.

Where ‘X bar’ represents the sample mean and n is the sample size.

Then, the variance of the new distribution is this  and the standard deviation is this.

If we have an estimate for a population mean, we can test it out using hypothesis testing.

The logic goes: if the estimate is correct, we’re very unlikely to observe some sample means. 

So if we take a sample and do observe those unlikely means, we should reject the estimate we currently have.  

To find critical regions on a normal distribution, use the inverse normal distribution function on your calculator. 

The hypotheses look pretty much the same as when we test using a binomial distribution.

It’s just that, since we’re forming hypotheses about mean values now, we use.

Finally then, the exam may tell you a sample was taken, tell you the significance level and ask you to perform the rest of the hypothesis test. 

Here’s what that looks like for a one-tailed test.

First, write down the null and alternative hypotheses.

Second, write down the distribution for the sample mean.

Third, find and state the critical region.

Fourth, compare the test statistic to the critical region.

And fifth, write a conclusion.

An alternative method is to find the probability of observing the sample mean or anything greater,

Compare that to the significance level, and then write the conclusion accordingly.

The working looks very similar for a two-tailed test.

First, write down the null and alternative hypotheses.

Second, write down the distribution for the sample mean.

Third, find and state the critical region, now with two tails.

Fourth, compare the test statistic to the critical region.

And fifth, write a conclusion. 

In place of this step, you could find the probability of observing the sample mean or anything greater,

Compare that to half of the significance level, since this is a two-tailed test, 

And then write the conclusion accordingly.