Comparing P-value from t statistic to significance level | AP Statistics | Khan Academy

Jude was curious if the automated machine at his restaurant was filling drinks with the proper amount. He filled a sample of 20 drinks to test his null hypothesis, which is the actual population mean for how much drink there was in the drinks per drink is 530 milliliters versus his alternative hypothesis, which is that the population mean is not 530 milliliters. Where mu is the mean filling amount.

The drinks in the sample contained a mean amount of 528 milliliters with a standard deviation of 4 milliliters. These results produced a test statistic of t is equal to negative 2.236 and a p-value of approximately 0.038. Assuming the conditions for inference were met, what is an appropriate conclusion at the alpha equals 0.05 significance level?

They give us some choices here, and like always, I encourage you to pause this video and see if you can figure it out on your own.

All right, so now let's work through this together. So let's just remind ourselves what's going on. You have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters. But then there's the alternative hypothesis that it's not 530 milliliters.

So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks, and using that sample, you calculate a sample mean and then you also calculate a sample standard deviation. They tell us these things right over here. Using this information and actually our sample size, you are able to calculate a t statistic.

You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. The p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true. If that probability is lower than our significance level, then we say, "Hey, that's a very low probability. We're going to reject our null hypothesis," which would suggest our alternative.

So the key to this question is just to compare this p-value right over here to our significance level. As we see, the p-value 0.038 is indeed less than 0.05, and so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative that the true mean is something different than 530 milliliters.

If we look at our choices here, the first choice says, "Reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 milliliters." Yeah, that one looks good. This suggests this is strong evidence; this suggests the alternative hypothesis, which is that right over there.

Let's read the other ones just to make sure that they don't make sense. So this is rejecting the null hypothesis; that looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 milliliters. No, not the first one is definitely much stronger.

Fail to reject the null hypothesis? No, we are rejecting the null hypothesis because our p-value is lower than our significance level. Fail to reject? Now we'd rule that one out as well.