Question

Given a linear correlation coefficient r = 0.476, a sample size n = 20, and a significance level of α = 0.05, use Table A-6 to determine the critical value of r and state if the given r represents a significant linear correlation. Would your answer change if the significance level was α = 0.01?

Answer 1

CASE 1 - When α = 0.05

Let ρ = population correlation coefficient (unknown)
r = sample correlation coefficient (known; calculated from sample data)

Null and Alternative hypothesis:
H0​:ρ=0 i.e. population correlation coefficient is 'close to 0'
HA​:ρ≠0 i.e. population correlation coefficient is 'significantly different from 0'

Value of correlation coefficient:
r=0.476

Degrees of freedom
The sample size is n=20, so then the number of degrees of freedom is df = n-2 = 20 - 2 = 18

                                                           Table of Critical Values Method
Critical region:
The corresponding critical correlation value rc​ for a significance level of α=0.05, for a Two-tailed test is: rc=0.4438

Rejection Region
Observe that in this case, the null hypothesis H0​:ρ=0 is rejected if |r|>rc=0.4438.

Decision about the null hypothesis
Based on the sample correlation provided, we have that |r|=0.476>rc​=0.4438, from which is concluded that the null hypothesis is rejected.

CASE 2 - When α = 0.01

Let ρ = population correlation coefficient (unknown)
r = sample correlation coefficient (known; calculated from sample data)

Null and Alternative hypothesis:
H0​:ρ=0 i.e. population correlation coefficient is 'close to 0'
HA​:ρ≠0 i.e. population correlation coefficient is 'significantly different from 0'

Value of correlation coefficient:
r=0.476

Degrees of freedom
The sample size is n=20, so then the number of degrees of freedom is df = n-2 = 20 - 2 = 18

                                                           Table of Critical Values Method
Critical region:
The corresponding critical correlation value rc​ for a significance level of α=0.01, for a Two-tailed test is: rc=0.5614

Rejection Region
Observe that in this case, the null hypothesis H0​:ρ=0 is rejected if |r|>rc=0.5614.

Decision about the null hypothesis
Based on the sample correlation provided, we have that |r|=0.476<rc​=0.5614, from which is concluded that the null hypothesis is NOT rejected.

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