Question

In this problem, we use your critical values table to explore the significance of r based on different sample sizes.

(a) Is a sample correlation coefficient ρ = 0.84 significant at the α = 0.01 level based on a sample size of n = 5 data pairs? What about n = 9 data pairs? (Select all that apply.)

No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 9 and α = 0.01.

Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 5 and α = 0.01.

Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 9 and α = 0.01.

No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 9 and α = 0.01.

No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 5 and α = 0.01.

Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 9 and α = 0.01.

Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 5 and α = 0.01.

No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 5 and α = 0.01.

(b) Is a sample correlation coefficient ρ = 0.41 significant at the α = 0.05 level based on a sample size of n = 16 data pairs? What about n = 29 data pairs? (Select all that apply.)

Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 29 and α = 0.05.

Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 16 and α = 0.05.

No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 29 and α = 0.05.

No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 29 and α = 0.05.

Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 29 and α = 0.05.

No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 16 and α = 0.05.

No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 16 and α = 0.05.

Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 16 and α = 0.05.

Answer 1

we want to test the significance of r.

The null & alternative Hypothesis:

Ho:- r is not significant.

vs

H1:- r is significant.

we will find critical value from the table for df = n-2.

If the r-value is greater than the positive critical value then we can conclude that the correlation coefficient is significant.

1)  Is a sample correlation coefficient ρ = 0.84 significant at the α = 0.01 level based on a sample size of n = 5 data pairs? What about n = 9 data pairs?

i) for ρ = 0.84 & the α = 0.01 level based on a sample size of n = 5.

df = n-2 = 5-2 = 3

for df = 3 ;  α = 0.01

The critical value from table is 0.958735

ρ = 0.84 < 0.958735

correlation coefficient is not significant at for n= 5;  α = 0.01

ii)

for ρ = 0.84 & the α = 0.01 level based on a sample size of n = 9.

df = n-2 = 9-2 = 7

for df = 7;  α = 0.01

The critical value from table is 0.797681

ρ = 0.84 >  0.797681

correlation coefficient is significant at for n= 9;  α = 0.01

Answer:-

i) Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 9 and α = 0.01.

ii) No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 5 and α = 0.01.

2)

Is a sample correlation coefficient ρ = 0.41 significant at the α = 0.05 level based on a sample size of n = 16 data pairs? What about n = 29 data pairs?

i) for ρ = 0.41 & the α = 0.05 level based on a sample size of n = 16.

df = n-2 = 16-2 = 14

for df = 14;  α = 0.05

The critical value from table is

ρ = 0.41 >  0.497309

correlation coefficient is significant at for n= 16;  α = 0.05

ii)

for ρ = 0.41 & the α = 0.05 level based on a sample size of n = 29.

df = n-2 = 29-2 = 27

for df = 27;  α = 0.05

The critical value from table is 0.367278

ρ = 0.41 < 0.367278

correlation coefficient is not significant at for n= 29;  α = 0.05

Answer:-

i) Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 16 and α = 0.05.

ii)No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 29 and α = 0.05.