R. A. Fisher has derived the sampling distribution of the correlation coefficient defined in Eq. (3.5.13). xy

R. A. Fisher has derived the sampling distribution of the correlation coefficient defined in Eq. (3.5.13).

If it is assumed that the variables X and Y are jointly normally distributed, that is, if they come from a bivariate normal distribution , then under the assumption that the population correlation coefficient ρ is zero, it can be shown that t = r ˆš(n €“ 2)/ ˆš (1 ˆ’ r²) follows Student€™s t distribution with n ˆ’ 2 df.** Show that this t value is identical with the t value given in Eq. (5.3.2) under the null hypothesis that β₂ = 0. Hence establish that under the same null hypothesis F = t².

Eq (5.3.2) : t = (β̂₂ €“ β₂) ˆšÎ£x_i² / σ̂

Transcribed Image Text:

Σxy / (Σ) (ΣΥ) η ΣΧΥ - (Σ Χ)(ΣΥ) Vi"ΣΧ- (Σ Χ);]μ ΣΥ -(Σ]