Using TI calculator for P-value from t statistic | AP Statistics | Khan Academy

Miriam was testing her null hypothesis that the population mean of some dataset is equal to 18 versus her alternative hypothesis that the mean is less than 18. With a sample of 7 observations, her test statistic, I can never say that was T, is equal to negative 1.9. Assume that the conditions for inference were met. What is the approximate p-value for Miriam's test?

So pause this video and see if you can figure this out on your own.

All right, well, I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18.

To test that null hypothesis, Miriam takes a sample; sample size is equal to 7. From that, she would calculate her sample mean and her sample standard deviation, and from that, she would calculate this T statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they'd say, okay, well, we'd say the T statistic is equal to her sample mean minus the assumed mean from the null hypothesis.

That's what we have over here, divided by—and this is a mouthful—our approximation of the standard error of the mean. The way we get that approximation is we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us; this is equal to negative 1.9.

If we think about a T distribution, I'll try to hand draw a rough T distribution really fast. If this is the mean of the T distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, is: well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9, so it's this area right there.

I'm going to do this with a TI-84, at least an emulator of a TI-84, and all we have to do is go to second distribution. Then I would use the T cumulative distribution function. So let's go there. That's the number six right there; click enter.

So my lower bound, yeah, I essentially wanted it to be negative infinity. So we can just call that negative infinity as just an approximation of negative infinity—a very, very low number. Our upper bound would be negative 1.9. Negative 1.9.

Then our degrees of freedom—that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six. And so there we have it. This would be our p-value, which would be approximately 0.053.

So our p-value would be approximately 0.053. Then what Miriam would do is compare this p-value to her preset significance level, alpha. If this is below alpha, then she would reject her null hypothesis, which would suggest the alternative. If this is above alpha, then she would fail to reject her null hypothesis.