A sports statistician investigated the relationship between a professional basketball player's salary, in hundreds of thousands of dollars, and the percentage of shots, in decimal form, that she made. The statistician performed a linear regression on the data with a graphing calculator. The results are shown. LinReg y = ax+b a = .2769230769 b = .3884615385 r2 = .8613966678 r = .9281145769 What conclusion can be made from the data? Select from the drop-down menus to correctly complete the sentence. Rounded to the nearest hundredth, the correlation coefficient is Choose... so there Choose... a strong correlation between salary and percentage of shots made.
The Correct Answer and Explanation is:
Correct Answer:
Rounded to the nearest hundredth, the correlation coefficient is 0.93, so there is a strong correlation between salary and percentage of shots made.
Explanation (300 words):
The correlation coefficient, often represented as r, measures the strength and direction of the linear relationship between two variables. Its value ranges from -1 to 1. A value close to 1 indicates a strong positive linear relationship, a value close to -1 indicates a strong negative linear relationship, and a value near 0 suggests no linear relationship.
In this case, the statistician obtained r = 0.9281145769, which rounds to 0.93 when taken to the nearest hundredth. This value lies very close to 1, which indicates a strong positive correlation.
A strong positive correlation means that as one variable increases, the other tends to increase as well. Here, the two variables being compared are a basketball player’s salary (in hundreds of thousands of dollars) and the percentage of shots made (in decimal form). A high correlation coefficient like 0.93 implies that players who make a higher percentage of their shots tend to earn higher salaries. Conversely, those with lower shooting percentages tend to earn less.
The linear regression equation provided, y = 0.2769230769x + 0.3884615385, further supports this interpretation. The positive slope (a = 0.2769) suggests that for each unit increase in shooting percentage (measured in decimal form), the predicted salary increases.
Therefore, the data shows a clear trend: better shooting accuracy is associated with higher salaries in professional basketball. The strength of the correlation suggests that shooting performance is a significant factor in determining pay, although other factors like experience, defensive skill, and teamwork may also influence salary but are not included in this analysis.
