Introduction to t statistics | Confidence intervals | AP Statistics | Khan Academy

We have already seen a situation multiple times where there is some parameter associated with the population. Maybe it's the proportion of a population that supports a candidate; maybe it's the mean of a population, the mean height of all the people in the city.

We've determined that it's unpractical or we just—there's no way for us to know the true population parameter. But we could try to estimate it by taking a sample size. So, we take n samples and then we calculate a statistic based on that.

We've also seen that not only can we calculate the statistic, which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. That confidence interval would look something like this: it would be the value of the statistic that we have just calculated plus or minus some margin of error.

We’ll often say this critical value, z, and this will be based on the number of standard deviations we want to go above and below that statistic. Then, we'll multiply that times the standard deviation of the sampling distribution for that statistic.

Now, what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is, say, the population proportion—what percentage of the population supports a certain candidate?

Well, in that world the statistic is the sample proportion. So, we would have the sample proportion plus or minus z star times—well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times 1 minus p hat, the sample proportion times 1 minus the sample proportion over our sample size.

If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic; our sample mean plus or minus z star. Now, if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size.

But we often will not know this. In fact, it's very unusual to know this. So sometimes you will say, "Okay, if we don't know this, let's just figure out the sample standard deviation of our sample." Here, instead we'll say, "Okay, let's take our sample mean plus or minus z star times the sample standard deviation of our sample, which we can calculate divided by the square root of n."

Now, this might seem pretty good if we're trying to construct a confidence interval for our sample for our mean, but it turns out that this is not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level.

And so that's why statisticians have invented another statistic. Instead of using z, they call it t. Instead of using a z table, they use a t table, and we're going to see this in future videos.

So if you are actually trying to construct a confidence interval for a sample mean, and you don't know the true standard deviation of your population—which is normally the case—instead of doing this, what we're going to do is we're going to take our sample mean plus or minus our critical value. We'll call that t star times our sample standard deviation, which we can calculate divided by the square root of n.

So the real functional difference is that this actually is going to give us the confidence interval that actually has the level of confidence that we want. If we have 195 percent level of confidence, if we keep computing this over and over again for multiple samples, that roughly 95 percent of the time this interval will contain our true population mean.

To functionally do it—and we'll do it in future videos—you really just have to look up a t table instead of a z table.