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 = 0.82 significant at the α = 0.01 level based on a sample size of n = 3 data pairs? What about n = 14 data pairs? (Select all that apply.) No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 14 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 = 3 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 = 3 and α = 0.01. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 3 and α = 0.01. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 14 and α = 0.01. No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 3 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 = 14 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 = 14 and α = 0.01. Incorrect: Your answer is incorrect. (b) Is a sample correlation coefficient ρ = 0.42 significant at the α = 0.05 level based on a sample size of n = 18 data pairs? What about n = 26 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 = 26 and α = 0.05. No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 18 and α = 0.05. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 18 and α = 0.05. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 26 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 = 26 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 = 18 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 = 18 and α = 0.05. No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 26 and α = 0.05. Incorrect: Your answer is incorrect. (c) Is it true that in order to be significant, a ρ value must be larger than 0.90? larger than 0.70? larger than 0.50? What does sample size have to do with the significance of ρ? Explain your answer. No, a larger sample size means that a smaller absolute value of the correlation coefficient might be significant. No, sample size has no bearing on whether or not the correlation coefficient might be significant. Yes, a larger correlation coefficient of 0.70 means that the data will be significant. Yes, a larger correlation coefficient of 0.90 means that the data will be significant. Yes, a larger correlation coefficient of 0.50 means that the data will be significant.

Answer 1

We are asked to perform correlation test .

H0 : = 0 , Ha :   ≠0  

Decision rule :

Reject H0, If absolute value of correlation coefficient is greater than or equal to ( ≥) critical value, so there is significant correlation.

Fail to reject H0, if absolute value of correlation coefficeint is less than (<) critical value , so there is no significant correlation.

a) We are given sample correlation coefficient r = 0.82 and n = 3 ,n =14 and = 0.01

So using above critical value table for = 0.01 ,

For n = 3 , critical value = 1

As | r |< critical value ,we fail to reject H0,so there is no significant correlation.

For n =14 , critical value = 0.66

As | r | > critical value , we reject H0, so there is significant correlation.

Answer :

No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 3 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 = 14 and α = 0.01.

B) r = 0.42, = 0.05

For n = 18, critical value = 0.47

As | r | < critical value , we fail to reject H0, there is no significant correlation.

For n = 26 , critical value = 0.39

As | r | > critical value , we reject H0, so there is significant correlation.

Answer :

Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 26 and α = 0.05.

No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 18 and α = 0.05.

C)

We can see on the table , as sample size increases , critical value decreases ,So for larger sample size the smaller absolute value of sample correlation coefficient might be significant.

Answer : No, a larger sample size means that a smaller absolute value of the correlation coefficient might be significant.