Question

compute the linear correlation coefficient between the two variables, rounding to three decimal places, and determine whether a linear relation exists.

Answer 1

To achieve a coefficient of determination of at least 0.50, the correlation coefficient between two variables must be at least 0.707 or at most -0.707 when rounded to two decimal places.

Step-by-step explanation:

The question relates to the calculation of the linear correlation coefficient which is a statistical measure of the strength of a linear relationship between two variables. The coefficient of determination, represented as r², indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. To have a coefficient of determination of at least 0.50, the absolute value of the correlation coefficient, r, needs to be at least √0.50, which is approximately 0.707. This means the correlation must be at least 0.707 or at most -0.707 to reach the desired level of determination.

Answer 2

To answer the student's question, the absolute value of the correlation coefficient must be at least 0.707 to have a coefficient of determination of at least 0.50. The process involves using statistical methods to compute r and assess its significance.

Step-by-step explanation:

The question asks how to compute the linear correlation coefficient between two variables and assess the strength of the linear relationship. The linear correlation coefficient, commonly represented by the symbol r, measures the strength and direction of a linear relationship between two variables on a scatter plot. The closer the value of r is to 1 or -1, the stronger the linear relationship. A coefficient of determination, represented by r², indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. To have a coefficient of determination of at least 0.50, the absolute value of the correlation coefficient must be at least √0.50, which is approximately 0.707, rounded to three decimal places.

The steps often involved in such an analysis include plotting the data on a scatter plot, calculating the least-squares line (commonly the form ý = a + bx), drawing the line on the scatter plot, and then finding the correlation coefficient and assessing its significance. Significance is typically judged by using a hypothesis test or referring to critical values from a table based on the degrees of freedom. If the absolute value of r is greater than the critical value, the relationship is considered significant. For example, if r = 0.801 and the critical value is 0.632, then r is significant, and the regression line may be used for prediction.