How likely is it that a given sample correlation coefficient value (0.81 in this case) could arise by chance? Let's do a simulation involving pure chance. We'll do 10,000 correlations using nothing but purely random numbers. We'll use the same sample size as the *statistical scenario* and correlate 120 pairs of random numbers 10,000 separate times. Figure 38.1 is the histogram of simulation results. Notice that it is centered on zero and has a normal bell shape. This shows us what to expect when sampling from a population with a *population correlation equal to zero*, which is the null hypothesis.^{1}

You can see that sample correlations less than −0.2 or greater than +0.2 are unlikely to happen by chance with a sample size of 120 number pairs. Our *statistical scenario* sample correlation of 0.81 is almost impossible by chance. Its -value will be *extraordinarily* small.

What if we had a smaller sample size, say 30 number pairs? With smaller sample sizes, more extreme correlations should ...

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