The last application of the t-test on IB exams is testing whether there exists a correlation between two normally distributed random variables. If we have a sample of points \left(x,y\right) from X\sim\mathrm{N}\left(\mu_{x},\sigma_{x}^2\right), Y\sim\mathrm{N}\left(\mu_{y},\sigma_{y}^2\right), we can calculate the correlation coefficient r. A strong correlation in a large enough sample is strong evidence that the populations are correlated.

<ul>
<li>Two side-by-side scatter plots on a black background, each with white x and y axis labels.</li>
<li>In both plots, an orange dashed line slopes upward (positive slope).</li>
<li>Left plot:
<ul>
<li>Many small purple points scattered with moderate spread around the dashed line.</li>
<li>Several larger orange points positioned on or near the dashed line.</li>
<li>Orange/white label at the top right: r = 0.997.</li>
</ul>
</li>
<li>Right plot:
<ul>
<li>A larger number of small purple points, more densely clustered around the dashed line than in the left plot.</li>
<li>Several larger orange points along or near the dashed line.</li>
<li>Orange/white label at the top right: r = 0.879.</li>
<li>Purple annotation at the bottom right: “smaller r, but stronger evidence.”</li>
</ul>
</li>
</ul>

Basically, calculating r from a sample alone does not indicate how confident we can be that the underlying populations are correlated. If a small sample of \left(x,y\right) seems correlated, it could just be a coincidence.

This sounds abstract, so here is an example. The brightness of a star is correlated with the temperature at the surface of the star. By looking at a large enough number of stars, we can confidently conclude the population (all start in a certain temperature range) are correlated.

Note: This is not real data, but illustrates a real correlation.

We call the true correlation between the populations \rho, and r is only an estimate of \rho. Technically, r is a biased estimate of \rho, but all you need to know is that as the sample size goes to infinity, r\to\rho.

What we want is a test that combines the strength of the sample correlation r with the size of the sample. This is what the LinRegTTest on your calculator does.

Dr. Z	71	25	60	45	39	78
Mr. X	54	52	53	60	45	54

Dr. Z	71	25	60	45	39	78
Mr. X	54	52	53	60	45	54

Student's t-test

Student's t-test