3.3.2 Introducing the $t$ distribution

The second reason we previously required a large sample size was so that we could accurately estimate the standard error using the sample data. In the cases where we will use a small sample to calculate the standard error, it will be useful to rely on a new distribution for inference calculations: the $t$ distribution. A $t$ distribution, shown as a solid line in Figure LABEL:tDistCompareToNormalDist, has a bell shape. However, its tails are thicker than the normal model’s. This means observations are more likely to fall beyond two standard deviations from the mean than under the normal distribution.³²³²The standard deviation of the $t$ distribution is actually a little more than 1. However, it is useful to always think of the $t$ distribution as having a standard deviation of 1 in all of our applications. These extra thick tails are exactly the correction we need to resolve the problem of a poorly estimated standard error.

R> X <- seq(-5, 5, 0.01);Y <- dnorm(X)
R> plot(X, Y, type=’l’, lty=3,lwd=2.5)
R> Y <- dt(X, 2)
R> lines(X, Y, lwd=1.8, col=’blue’)

The $t$ distribution, always centred at zero, has a single parameter: degrees of freedom. The degrees of freedom (df) describe the precise form of the bell-shaped $t$ distribution. Several $t$ distributions are shown in Figure LABEL:tDistConvergeToNormalDist. When there are more degrees of freedom, the $t$ distribution looks very much like the standard normal distribution.

See the Moodle file for the code for the simulation.

When the degrees of freedom is about 30 or more, the $t$ distribution is nearly indistinguishable from the normal distribution. In Section 3.3.3, we relate degrees of freedom to sample size.

We will find it very useful to become familiar with the $t$ distribution, because it plays a very similar role to the normal distribution during inference for small samples of numerical data. We use qt function, in place of the qnorm function and pt instead of pnorm for small sample numerical data. The main difference is that there is no standard $t$ distribution and so we always need to specify the degrees of freedom as well as our quantile or probability of interest.

Answer. Just like for the normal problems, we first draw the picture in Figure LABEL:tDistDF18LeftTail2Point10 and shade the area below -2.10. To find this area, we identify the number of degrees of freedom: $df=18$. Then we do pt(-2.10,df=18) = in R.

Answer. We identify the degrees of freedom: $df=20$. Then we use R remembering we are looking for the upper tail: 1-pt(1.65,df=20)=.

Answer. As before, first identify the appropriate degrees of freedom: $df=2$. Next, we use R to find the upper tail 1-pt(3,df=2). Finally, we recall we want (symmetric) two tails (above and below 3) so we double the answer: 2*(1-pt(3,df=2))=0.09546597.

Answer. We find the shaded area above -1.79 (we leave the picture to you).

1-pt(-1.79,df=19).