Question

Finding Critical r Values Table A-6 lists critical values of r for selected values of n and a. More generally, critical r values can be found by using the formula

\(r = \frac{t}{{\sqrt {{t^2} + n - 2} }}\)

where the t value is found from the table of critical t values (Table A-3) assuming a two-tailed case with n - 2 degrees of freedom. Use the formula for r given here and in Table A-3 (with n - 2 degrees of freedom) to find the critical r values corresponding to \({H_1}:\rho \ne 0\), \(\alpha \)= 0.02, and n = 27.

Short answer

The critical values of r calculated using the given formula are -0.445 and 0.445.

Step-by-step solution

Given information

The critical value of r needs to be computed using the given formula.

The degrees of freedom are given to be equal to n-2. The sample size (n) is equal to 27 and the level of significance \(\left( \alpha \right)\) is equal to 0.02. \(\)

Computation of the critical value

The critical value can be computed using the formula given below.

\(r = \frac{t}{{\sqrt {{t^2} + n - 2} }}\)

Where

t is the critical value obtained from t distribution table with n-2 degrees of freedom and\(\alpha = 0.02\)

n is the sample size and has a value equal to 27.

Referring to the t distribution table, the critical value of t with 25 degrees of freedom for 0.02 significance level is 2.4851.

The critical value is computed as shown below:

\(\begin{aligned} r &= \frac{t}{{\sqrt {{t^2} + n - 2} }}\\ &= \frac{{2.4851}}{{\sqrt {{{\left( {2.4851} \right)}^2} + 27 - 2} }}\\ &= 0.445\end{aligned}\)

Thus, the critical r values are -0.445 and 0.445.