Question

Determine the critical values that would be used in testing each of the following null hypotheses using the classical approach. (Give your answers correct to three decimal places.)

(a) H_o: ρ = 0 vs. H_a: ρ ≠ 0, with n = 18 and α = 0.05
(smaller value)
(larger value)

(b) H_o: ρ = 0 vs. H_a: ρ > 0, with n = 32 and α = 0.01


(c) H_o: ρ = 0 vs. H_a: ρ < 0, with n = 16 and α = 0.05

Answer 1

Solution:

We have to determine the critical values that would be used in testing the population correlation coefficient ρ .

Part a) H_o: ρ = 0 vs. H_a: ρ ≠ 0, with n = 18 and α = 0.05

Since Ha is not equal to ( ≠ ) type, this is two tailed test.

For testing correlation coefficient , df = degrees of freedom are given by:

df = n - 2

thus

df = 18 - 2

df = 16

Two tail area =  α = 0.05

Look in t table for df = 16 and Two tail area =  α = 0.05 and find corresponding t critical values.

t critical value = 2.120

Since this is two tailed test, we have two critical values.

Smaller value = - 2.120

Larger value = 2.120

.

.

Part b) H_o: ρ = 0 vs. H_a: ρ > 0, with n = 32 and α = 0.01

Ha is > type, thus this is right tailed ( one tailed) test.

df = n - 2

df = 32 - 2

df = 30

One tail area =  α = 0.01

Look in t table for df = 30 and one tail area =  α = 0.01 and find corresponding t critical value.

t critical value = 2.457

.

.

Part c) H_o: ρ = 0 vs. H_a: ρ < 0, with n = 16 and α = 0.05

Ha is < type, thus this is left tailed ( one tailed) test.

df = n - 2

df =16 - 2

df = 14

One tail area= α = 0.05

.Look in t table for df = 14 and one tail area =  α = 0.05 and find corresponding t critical value.

t critical value = 1.761

Since this is left tailed test, t critical value is negative,

thus t critical value = - 1.761