Observation: Property 1 can be used to test the hypothesis that population random variables x and y are independent i.e. Observation: If we solve the equation in Property 1 for r, we get Here the numerator r of the random variable t is the estimate of ρ = 0 and s ris the standard error of r. If x and y have a bivariate normal distribution or if the sample size n is sufficiently large, then r has a normal distribution with mean 0, and t = r/s r ~ T( n – 2) where If ρ ≠ 0, then the sampling distribution is asymmetric and so the following property does not apply, and other methods of inference must be used. The sampling distribution of r is only symmetric when ρ = 0 (i.e. You can think of a bivariate normal distribution as the three-dimensional version of the normal distribution, in which any vertical slice through the surface which graphs the distribution results in an ordinary bell curve.
This time we require that x and y have a joint bivariate normal distribution or that samples are sufficiently large. The following property is analogous to the Central Limit Theorem, but for r instead of x̄. As in Sampling Distributions, we can consider the distribution of r over repeated samples of x and y.
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