Example constructing a t interval for a mean | Confidence intervals | AP Statistics | Khan Academy

A nutritionist wants to estimate the average caloric content of the burritos at a popular restaurant. They obtain a random sample of 14 burritos and measure their caloric content. Their sample data are roughly symmetric, with a mean of 700 calories and a standard deviation of 50 calories. Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos?

So pause this video and see if you can figure it out.

All right, what's going on here? So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but can't; doesn’t know the true population parameter here, the population mean. They take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. They want to use this data to construct a 95 percent confidence interval.

Our confidence interval is going to take the form, and we've seen this before: our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. If we used σ (sigma), which is a population parameter, then we could use a z-statistic right over here; we would use a z distribution.

But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus. So what would be our critical value for a 95 percent confidence interval? Well, we will just get out our t-table.

Remember, with a t-table, you have to care about degrees of freedom. If our sample size is 14, then that means you take 14 minus 1. So degrees of freedom is n minus 1. So that's going to be 14 minus 1, which is equal to 13. We have 13 degrees of freedom that we have to keep in mind when we look at our t-table.

So let's look at our t-table. For a 95 percent confidence interval and 13 degrees of freedom, degrees of freedom right over here, we have 13 degrees of freedom. That is this row right over here. If we want a 95 percent confidence level then that means our tail probability—remember if our distribution, let me see if I'll draw it really small—a little small distribution right over here. So if you want 95 of the area in the middle, that means you have 5 percent not shaded in, and that's evenly divided on each side.

So that means you have 2.5 percent at the tails. You want to look for a tail probability of 2.5 percent. That is this right over here: 0.025. That's two and a half percent. There you go, that is our critical value: 2.160.

So this part right over here, this is going to be 2.16 times—what's our sample standard deviation? It's 50 over the square root of n, square root of 14. All of our choices have the 700 there, so we just need to figure out what our margin of error, this part of it, and we could use a calculator for that.

Okay, 2.16, I could write a 0 there, doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here. I think 28.86, so this part right over here is approximately 28.86. That's our margin of error.

We see out of all of these choices here, if we round to the nearest tenth, that would be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one. I guess they're trying to make sure that we're looking at enough digits.

So there we have it. We have established our 95 percent confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval?

Here we have to think, well, did we take a random sample? They tell us that they obtained a random sample of 14 burritos, so we check that one.

Is the sampling distribution roughly normal? If we take over 30 samples then it would be, but here we only took 14. However, they do tell us that the sample data is roughly symmetric. So if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal.

Then the last condition is the independence condition. Here, if we aren't sampling with replacement—and it doesn't look like we are—if we're not sampling with replacement, this has to be less than 10 percent of the population of burritos; we're assuming that there's going to be more than 140 burritos that the universe, that the population, that this popular restaurant makes.

So I think we can meet the independence condition as well. Assuming that you feel good about constructing a confidence interval, this is the one that you would actually construct.