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LOS c: Formulate a test of the hypothesis that the population correlation coefficient equals zero and determine whether the hypothesis is rejected at a given level of significance.

We are examining the relationship between the number of cold calls a broker makes and the number of accounts the firm as a whole opens. We have determined that the correlation coefficient is equal to 0.70, based on a sample of 16 observations. Is the relationship statistically significant at a 10% level of significance, why or why not? The relationship is:

A)	significant; the t-statistic exceeds the critical value by 1.91.

B)	significant; the t-statistic exceeds the critical value by 3.67.

C)	not significant; the critical value exceeds the t-statistic by 1.91.

A)	significant; the t-statistic exceeds the critical value by 1.91.

B)	significant; the t-statistic exceeds the critical value by 3.67.

C)	not significant; the critical value exceeds the t-statistic by 1.91.

The calculated test statistic is t-distributed with n – 2 degrees of freedom:

t = r√(n – 2) / √(1 – r²) = 2.6192 / 0.7141 = 3.6678

From a table, the critical value = 1.76

A study of 40 men finds that their job satisfaction and marital satisfaction scores have a correlation coefficient of 0.52. At 5% level of significance, is the correlation coefficient significantly different from 0?

H₀: r = 0 vs. H_a: r ≠ 0

t = [r √(n – 2)] / √(1 – r2)

t_c (α = 0.05 and degrees of freedom = 38) = 2.021

t > t_c hence we reject H₀.

Suppose the covariance between Y and X is 0.03 and that the variance of Y is 0.04 and the variance of X is 0.12. The sample size is 30. Using a 5% level of significance, which of the following is most accurate? The null hypothesis of:

A)	significant correlation is rejected.

B)	no correlation is not rejected.

C)	no correlation is rejected.

A)	significant correlation is rejected.

B)	no correlation is not rejected.

C)	no correlation is rejected.

The correlation coefficient is r = 0.03 / (√0.04 * √0.12) = 0.03 / (0.2000 * 0.3464) = 0.4330.

The test statistic is t = (0.4330 × √28) / √(1 ? 0.1875) = 2.2912 / 0.9014 = 2.54.

The critical t-values are ± 2.048. Therefore, we reject the null hypothesis of no correlation.

Consider a sample of 60 observations on variables X and Y in which the correlation is 0.42. If the level of significance is 5%, we:

A)	cannot test the significance of the correlation with this information.

B)	conclude that there is statistically significant correlation between X and Y.

C)	conclude that there is no significant correlation between X and Y.

Consider a sample of 60 observations on variables X and Y in which the correlation is 0.42. If the level of significance is 5%, we:

A)	cannot test the significance of the correlation with this information.

B)	conclude that there is statistically significant correlation between X and Y.

C)	conclude that there is no significant correlation between X and Y.

The calculated t is t = (0.42 × √58) / √(1-0.42^2) = 3.5246 and the critical t is approximately 2.000. Therefore, we reject the null hypothesis of no correlation.

Consider a sample of 32 observations on variables X and Y in which the correlation is 0.30. If the level of significance is 5%, we:

A)	conclude that there is significant correlation between X and Y.

B)	cannot test the significance of the correlation with this information.

C)	conclude that there is no significant correlation between X and Y.

Consider a sample of 32 observations on variables X and Y in which the correlation is 0.30. If the level of significance is 5%, we:

A)	conclude that there is significant correlation between X and Y.

B)	cannot test the significance of the correlation with this information.

C)	conclude that there is no significant correlation between X and Y.

The calculated t = (0.30 × √30) / √(1 ? 0.09) = 1.72251 and the critical t values are ± 2.042. Therefore, we fail to reject the null hypothesis of no correlation.

Suppose the covariance between Y and X is 10, the variance of Y is 25, and the variance of X is 64. The sample size is 30. Using a 5% level of significance, which of the following statements is most accurate? The null hypothesis of:

A)	no correlation is rejected.

B)	significant correlation is rejected.

C)	no correlation cannot be rejected.

A)	no correlation is rejected.

B)	significant correlation is rejected.

C)	no correlation cannot be rejected.

The correlation coefficient is r = 10 / (5 × 8) = 0.25. The test statistic is t = (0.25 × √28) / √(1 ? 0.0625) = 1.3663. The critical t-values are ± 2.048. Therefore, we cannot reject the null hypothesis of no correlation.