t-distribution

1. t-distribution

So far we have discussed inference on numerical data using simulation based methods. When the parameter of interest is the mean (or a difference between two means) and certain conditions are met, we can also use methods based on the central limit theorem for conducting inference.

2. t-distribution

In a nutshell the t-distribution is useful for describing the distribution of the sample mean when the population standard deviation, sigma, is unknown, which is almost always. This distribution also has a bell shape, so it's unimodal and symmetric. It looks a lot like the normal distribution, however its tails are thicker. Comparing the normal and t distributions visually is the best way to understand what we mean by "thick tails". Notice that the peak of the T distribution doesn't go as high as the peak of the normal distribution. In other words, the T distribution is somewhat squished in the middle, and the additional area is added to the tails. This means that under the t distribution observations are more likely to fall 2 standard deviations away from the mean than under the normal distribution. This implies that confidence intervals constructed using the t distribution will be wider, in other words more conservative, than those constructed with the normal distribution.

3. Shape of the t-distribution

The t distribution is always centered at zero, and it has one parameter, degrees of freedom, which determines the thickness of its tails. What happens to the shape of the t distribution as the degrees of freedom increases? This plot shows a series of bell curves, going from light to dark shades of gray as the degrees of freedom increases. We can see that as the degrees of freedom increases, the t distribution approaches the normal distribution. This is why you might see in other resources that if the sample size is large enough one can use the normal distribution to approximate the sampling distribution of the mean. Often such statements will be accompanied by a sample size that can be considered large enough for the statement to hold. In this course we will keep things simple and straightforward -- when working with the mean (or difference of means) use the t distribution for central limit theorem based inference.

4. Let's practice!

Now let's try some examples.

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