Question

11.4#10

(a) Suppose n = 6 and the sample correlation coefficient is r = 0.906. Is r significant at the 1% level of significance (based on a two-tailed test)? (Round your answers to three decimal places.)

t	=
critical t	=

Conclusion:

Yes, the correlation coefficient ? is significantly different from 0 at the 0.01 level of significance.No, the correlation coefficient ? is not significantly different from 0 at the 0.01 level of significance.    

(b) Suppose n = 10 and the sample correlation coefficient is r = 0.906. Is r significant at the 1% level of significance (based on a two-tailed test)? (Round your answers to three decimal places.)

t	=
critical t	=

Conclusion:

Yes, the correlation coefficient ? is significantly different from 0 at the 0.01 level of significance.No, the correlation coefficient ? is not significantly different from 0 at the 0.01 level of significance.    

(c) Explain why the test results of parts (a) and (b) are different even though the sample correlation coefficient r = 0.906 is the same in both parts. Does it appear that sample size plays an important role in determining the significance of a correlation coefficient? Explain.

As n increases, the degrees of freedom and the test statistic decrease. This produces a smaller P value.As n increases, so do the degrees of freedom, and the test statistic. This produces a larger P value.    As n decreases, the degrees of freedom and the test statistic increase. This produces a smaller P value.As n increases, so do the degrees of freedom, and the test statistic. This produces a smaller P value.

Answer 1

Hence we fail to reject null hypothesis and conclude that

No, the correlation coefficient is not significantly different from 0 at the 0.01 level of significance.

Hence we reject null hypothesis and conclude that

Yes, the correlation coefficient is significantly different from 0 at the 0.01 level of significance.

(c)

We observe that the test results of parts (a) and (b) are different even though the sample correlation coefficient r = 0.906 is the same in both parts. Because sample size plays an important role in determining the significance of a correlation coefficient. We explain it via following graph:

From the graphs it is observed that for n=10 the pickness is high than for n=6 and the tail probability is smaller when n=10 than n=6 as a result p-value is smaller for n=10 than for n=6. Moreover value of test statistic increases as n increases since value of test statistic directly proportional to n for fixed r (see (1). So, As n increases, so do the degrees of freedom, and the test statistic. This produces a smaller P value.