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Writing hypotheses for a significance test about a mean | AP Statistics | Khan Academy

2m read
·Nov 11, 2024

A quality controlled expert at a drink bottling factory took a random sample of bottles from a batch and measured the amount of liquid in each bottle in the sample. The amount in the sample had a mean of 503 milliliters and a standard deviation of 5 milliliters. They want to test if this is convincing evidence that the mean amount for bottles in this batch is different than the target value of 500 milliliters. Let mu be the mean amount of liquid in each bottle in the batch.

Write an appropriate set of hypotheses for their significance test for the significance test that the quality control expert is running. So pause this video and see if you can do that.

All right, now let's do this together. So first you're going to have two hypotheses. You're going to have your null hypothesis and your alternative hypothesis. Your null hypothesis is going to be a hypothesis about the population parameter that you care about, and it's going to assume kind of the status quo—no news here.

The parameter that we care about is the mean amount of liquid in the bottles in the batch, so that's mu right over there. What would be the assumption that that would be the no news here? Well, it would be 500 milliliters—that's the target value. So, it's reasonable to say, “Well, you know the null is it's doing what it's supposed to.” That where the actual mean for the batch is actually what the target needs to be is actually 500 milliliters.

Some of you might have said, “Hey, wait! Didn't they say the amount in the sample had a mean of 503 milliliters? Why isn't this 503?” Remember, your hypothesis is going to be about the population parameter—your assumption about the population parameter. This 503 milliliters right over here, this is a sample statistic; this is a sample mean that's trying to estimate this thing right over here.

When we do our significance test, we're going to incorporate this 503 milliliters. We're going to think about, well, what's the probability of getting a sample statistic, a sample mean, this far or further away from the assumed mean if we assume that the null hypothesis is true? If that probability is below a threshold—our significance level—then we reject the null hypothesis and it would suggest the alternative.

But if we're just trying to generate or write a set of hypotheses, this would be our null hypothesis. Then our alternative hypothesis is that the true mean for the batch is something different than 500 milliliters.

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