Question

maya records data showing the relationship between the number of hours she practices piano each week and the number of mistakes she makes during a recital. the correlation coefficient of the line of best fit is -0.90. a. are these variables correlated? explain your reasoning. b. does one variable cause the other to change? explain your reasoning.

Explanation:

a. Correlation coefficient \(r = - 0.90\). Since \(|r|=0.90>0.7\) (a common threshold for strong correlation), the variables are correlated. The negative sign indicates an inverse relationship.
b. Correlation does not imply causation. Just because there's a strong correlation (\(r=-0.90\)) between practice hours and mistakes, there could be other factors (e.g., quality of practice, prior skill level) influencing the number of mistakes.

Answer:

a. Yes, the variables are correlated. The correlation coefficient \(r=-0.90\) has a magnitude \(|r| = 0.90>0.7\) (a typical value for strong correlation), so there is a strong linear relationship.
b. No, one variable does not necessarily cause the other. Correlation (\(r =-0.90\)) shows a relationship, but there could be confounding variables (e.g., practice quality, initial skill) affecting the number of mistakes.