Q1. 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 3.67.

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

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

Q2. 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?

A)   No, t = 1.68.

B)   Yes, t = 3.76.

C)   No, t = 2.02.

Q3. 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)   no correlation is rejected.

B)   significant correlation is rejected.

C)   no correlation is not rejected.

Q4. 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.

Q5. 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)   conclude that there is no significant correlation between X and Y.

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

答案和详解如下：

Q1. 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 3.67.

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

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

Correct answer is C)

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

Q2. 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?

A)   No, t = 1.68.

B)   Yes, t = 3.76.

C)   No, t = 2.02.

Correct answer is B)

H₀: r = 0 vs. H_a: r ≠ 0

t = [r √(n – 2)] / √(1 – r2)

< 0.522)="3.76" – √(1 √(38)] >="[(0.52">

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

t > t_c hence we reject H₀.

Q3. 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)   no correlation is rejected.

B)   significant correlation is rejected.

C)   no correlation is not rejected.

Correct answer is A)

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.

Q4. 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.

Correct answer is B)

The calculated t is t = (0.42 × √58) / √0.8236 = 3.5246 and the critical t is approximately 2.000. Therefore, we reject the null hypothesis of no correlation.

Q5. 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)   conclude that there is no significant correlation between X and Y.

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

Correct answer is B)

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.

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