Question

If x and y in a regression model are totally unrelated, _______.

A) There is a perfect positive correlation.
B) There is a perfect negative correlation.
C) The correlation coefficient is zero.
D) It is impossible to determine the correlation.

Answer 1

The correlation coefficient is zero if variables x and y in a regression model are completely unrelated. This coefficient measures the strength of the linear relationship, which ranges from -1 to +1, with 0 signifying no relationship.

C) The correlation coefficient is zero.

Step-by-step explanation:

If x and y in a regression model are totally unrelated, the correct answer is C) The correlation coefficient is zero. The correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. This coefficient ranges from -1 to +1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship at all.

Conclusions about the significance of the correlation are often drawn from hypothesis tests. If the null hypothesis (that the correlation coefficient is equal to zero) is rejected in favor of the alternate hypothesis (Ha: The population correlation coefficient is significantly different from zero), then there is sufficient evidence to conclude a significant linear relationship exists between x and y.

In the case that x and y are totally unrelated, drawing a conclusion would lead to the statement: There is insufficient evidence to conclude there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.

Answer 2

If x and y in a regression model are totally unrelated, the correlation coefficient is zero (option C), reflecting no linear relationship between the variables. The correlation coefficient, r, which ranges from -1 to +1, is zero when there is no association, meaning changes in one variable do not predict changes in the other.

Step-by-step explanation:

If x and y in a regression model are totally unrelated, the correct answer is C) The correlation coefficient is zero. The correlation coefficient, often denoted as r, measures the strength and direction of a linear relationship between two variables on a scatterplot. If there is no relationship between the variables, we can expect the correlation coefficient to be zero.

The range of the correlation coefficient is from -1 to +1. A value of +1 indicates a perfect positive correlation, meaning that as one variable increases, the other also increases. A value of -1 indicates a perfect negative correlation, meaning that as one variable increases, the other decreases. A value of 0 indicates no relationship between the variables, implying that changes in one variable do not predict changes in the other. Therefore, if we conclude that variables x and y have a correlation coefficient significantly different from zero, we can say there is evidence of a significant linear relationship between them.

However, in this scenario where x and y are totally unrelated, we expect the correlation coefficient to not be significantly different from zero, indicating insufficient evidence of a linear relationship. This is further supported by the concept that a correlation coefficient closer to zero denotes a weaker relationship, making predictions less accurate. Hence, the alternate hypothesis Ha: The population correlation coefficient is not equal to zero would be rejected in this case.