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# Example calculating t statistic for a test about a mean

AP.STATS:
VAR‑7 (EU)
,
VAR‑7.E (LO)
,
VAR‑7.E.1 (EK)

## Video transcript

Rory suspects that teachers in his school district have less than five years of experience on average he decides to test his null hypothesis is 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 the 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 so 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 so if this represents all the teachers in the district the population then what he did is he took a sample and says he used a sample of 25 teachers so n here is equal to 25 and then from that sample he was able to calculate some statistics he was able to calculate the sample mean it's 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 test is we say all right if we assume the null hypothesis is true what's the probability of getting getting a sample mean this low or lower and 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 distance test statistic sometimes we use a z-test if we're using if we're dealing with proportions but when we deal with means we tend to use a t-test and the reason why is if you wanted to figure out a Z statistic what you would do is you would take your sample mean subtract from that the assumed mean from the null hypothesis Somu and i'll just put a little Zero Sub Zero there so 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 this we don't know and so that's why instead we do a t statistic in which case we take the difference between our sample mean and our sumed population mean the population parameter and we try to estimate this and we estimate that with our sample standard deviation divided by the square root of our sample size and so if you're inspired I encourage you I even if you're not inspired 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 4 minus our assumed mean is 5 our assumed population mean is 5 our sample standard deviation is 2 all of that over the square root of the sample size all of that over square root of 25 so this is going to be equal to our numerator is negative 1 so it's negative 1/2 over 5 which is equal to negative 1 times 5 over 2 and so this is going to be equal to equal to negative 5 over 2 or negative 2 point 5 and then what we would do in this what Rory would do is then 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 it's say okay we are negative 2.5 below the mean so negative negative 2.5 and so 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 and so that would give us our p value and then 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 we 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 that assume all conditions for inference have been met and so that's the assuming that this was truly a random sample and that each the individual observations are either truly independent or roughly independent then maybe he observed either with replacement or it's less than 10% of the population and he feels good that the sampling distribution is going to be roughly normal and we've talked about that in other videos