Rochelle builds a linear regression to help her estimate her monthly entertainment spending based on the number of hours she works each week. Let y be the amount of money she spends on entertainment, and let x be the number of hours she works each week. She develops a regression that states that y=500−3.975x. The regression has an r2 of 0.9867. What can be stated about the correlation between Rochelle's monthly entertainment spending and the number of hours she works each week?
A) The data have a weak positive correlation.
B) The data have a strong positive correlation.
C) The data have a strong negative correlation.
D) The data have a weak negative correlation.
To understand what can be stated about the correlation between Rochelle's monthly entertainment spending and the number of hours she works each week, we need to analyze the information provided.
Rochelle's regression equation is y = 500 - 3.975x, where y represents the amount of money spent on entertainment and x represents the number of hours worked per week.
To determine the correlation, we should look at the coefficient of determination, also known as r-squared (r²). In this case, it is stated that the regression has an r² value of 0.9867.
The coefficient of determination, r², measures the proportion of the variance in the dependent variable (y) that can be explained by the independent variable (x). It indicates how well the regression line fits the data. The value of r² ranges from 0 to 1, with 1 representing a perfect fit.
Given that the r² value is high (0.9867), it indicates that the regression line is able to explain approximately 98.67% of the variance in Rochelle's monthly entertainment spending based on the number of hours she works each week.
Since the r² value is positive, it suggests a positive correlation between the two variables. Therefore, the correct answer is:
B) The data have a strong positive correlation.