<p>My teacher never really taught us this so I have no idea what it is. I know that to test a null hypothesis you have to use this to prove that we can use a normal model, but how do you do that?</p>
<p>the CLT states that if the sample size is large enough (usually 40 or greater), the sample will have an approximately normal distribution even if the population is not normal. I don’t know how it works but it does lol =]</p>
<p>If you have a sample size n greater than 30 (at least, that’s how I learned it), then you can use the CLT to justify the assumption that the sample mean has an approximately normal distribution so that you can use a sample. (Our teacher hates it when we use the word “prove” in hypothesis testing problems, so try not to do that…)</p>
<p>You can also justify that assumption if the population mean has a normal distribution or by creating a box-and-whisker graph that shows that the distribution is symmetrical. If you can’t use either, you can only just say something to the effect of “we must assume that the distribution of the sample mean is normal”.</p>
<p>You need the central limit theorem because in a hypothesis test, you’re going to compute the probability of getting your sample or something less likely. In order to find this probability, you’ve got to know the distribution of your sample means. The central limit theorem says that if your sample size is “large”, the sample means distribution will look normal. The mathematical proof behind this is pretty intense (think graduate level!), but I think that it’s “believable”:</p>
<p>Let’s say that the average number of hours that students sleep at your high school is 6 per night. You randomly sample 50 students, and find the sample mean. You do this again and again and again. What would the histogram look like? Most of the sample means would be close to 6 (since it’s the true mean) and fewer and fewer would be more than 6 as well as less than 6. What’s the picture? A normal distribution!</p>
<p>I hope this helps</p>