Example calculating t statistic for a test about a mean | AP Statistics | Khan Academy

[Music] Rory suspects that teachers in his school district have less than five years of experience on average. He decides to test his null hypothesis: that the mean number of years of experience is five years, and his alternative hypothesis is that the true mean years of experience is less than five years.

Using a sample of 25 teachers, his sample mean was four years, and his sample standard deviation was two years. Rory wants to use these sample data to conduct a t-test on the mean. Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test.

So, I always just like to remind ourselves what's going on. You have your null hypothesis here that the mean number of years of experience for teachers in the district is five, and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district.

If this represents all the teachers in the district, the population, then what he did is he took a sample. It said he used a sample of 25 teachers, so n here is equal to 25. From that sample, he was able to calculate some statistics. He was able to calculate the sample mean; that sample mean was four years. The sample mean was four years, and then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years.

Now, the whole point that we do, or the main thing we do when we do significance testing, is we say, "All right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower?" If that probability is below a preset significance level, then we reject the null hypothesis, and it suggests the alternative.

But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test if we're dealing with proportions, but when we deal with means, we tend to use a t-test. The reason why is if you wanted to figure out a z statistic, what you would do is take your sample mean and subtract from that the assumed mean from the null hypothesis, so mu. I'll just put a little zero sub-zero there to show this is the assumed mean from the null hypothesis, and then you would want to divide by the standard deviation of the sampling distribution of the sample mean.

So, you'd want to divide by that, but we don't know. That's why instead we do a t statistic, in which we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. We estimate that with our sample standard deviation divided by the square root of our sample size.

If you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t statistic.

Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean, which is five. Our assumed population mean is five, our sample standard deviation is two, all of that over the square root of the sample size—all of that over the square root of 25.

So, this is going to be equal to our numerator, which is -1, divided by 2 over 5, which is equal to 1 times 5 over 2. Hence, this is going to be equal to -5 over 2, or -2.5.

Then what we would do in this—what Rory would do is look this t-value up on a t-table and say, "So if you look at a distribution of a t statistic something like that," and say, "Okay, we are -2.5 below the mean." So, -2.5.

What he would want to do is figure out this area here because this would be the probability of being that far below the mean or even further below the mean. That would give us our p-value. If that p-value is below some preset significance level that Rory should have set, maybe 5% or 1%, then he'll reject the null hypothesis, which would suggest his suspicion that the true mean of the years of experience for the teachers in this district is less than five.

Now, another really important thing to keep in mind is they told us to assume all conditions for inference have been met. That's the assumption that this was truly a random sample, that each individual observation is either truly independent or roughly independent—that maybe he observed either with replacement or it's less than 10% of the population. He feels good that the sampling distribution is going to be roughly normal, and we've talked about that in other videos.