Which of the following tests gives the same result as a test of the regression line slope in simple linear regression?

• A. A test for the Pearson correlation coefficient $r$ being significantly different from zero
• B. A test for the variance of the residuals
• C. A test for the mean of the response variable
• D. A test for the intercept of the regression line

Answer: (A)

Verified step by step guidance

1. Understand that in simple linear regression, the slope of the regression line measures the strength and direction of the linear relationship between the predictor variable (X) and the response variable (Y).
2. Recall that the Pearson correlation coefficient \( r \) quantifies the linear association between two variables, ranging from -1 to 1, and is related to the slope \( \beta_1 \) of the regression line by the formula \( \beta_1 = r \times \frac{s_Y}{s_X} \), where \( s_Y \) and \( s_X \) are the standard deviations of Y and X respectively.
3. Recognize that testing whether the slope \( \beta_1 \) is significantly different from zero is equivalent to testing whether the Pearson correlation coefficient \( r \) is significantly different from zero, because if there is no linear relationship, both \( \beta_1 \) and \( r \) will be zero.
4. Note that tests for the variance of residuals, the mean of the response variable, or the intercept of the regression line do not directly assess the linear relationship between X and Y, and thus do not give the same result as the slope test.
5. Therefore, conclude that the test for the Pearson correlation coefficient \( r \) being significantly different from zero gives the same result as the test for the slope of the regression line in simple linear regression.