A restaurant manager collected data to predict monthly sales for the restaurant from monthly advertising expenses. The model created from the data showed that 36 percent of the var... A restaurant manager collected data to predict monthly sales for the restaurant from monthly advertising expenses. The model created from the data showed that 36 percent of the variation in monthly sales could be explained by monthly advertising expenses. What was the value of the correlation coefficient?
Understand the Problem
The question is asking for the calculation of the correlation coefficient based on the given information that 36 percent of the variation in monthly sales can be explained by monthly advertising expenses. This requires applying the formula for the correlation coefficient, which is the square root of the coefficient of determination (R²). In this case, since R² = 0.36, we will find the square root of 0.36.
Answer
The correlation coefficient is \( r = 0.6 \).
Answer for screen readers
The value of the correlation coefficient is ( r = 0.6 ).
Steps to Solve
- Identify the Coefficient of Determination (R²)
From the problem, we know that the coefficient of determination, ( R^2 ), is given as 36 percent. We can express this as a decimal: $$ R^2 = 0.36 $$
- Calculate the Correlation Coefficient (r)
The correlation coefficient ( r ) can be found by taking the square root of ( R^2 ): $$ r = \sqrt{R^2} $$
So, substituting the value we found: $$ r = \sqrt{0.36} $$
- Perform the Calculation
Now we compute the square root: $$ r = \sqrt{0.36} = 0.6 $$
The value of the correlation coefficient is ( r = 0.6 ).
More Information
The correlation coefficient measures the strength and direction of a linear relationship between two variables. In this case, a correlation coefficient of 0.6 indicates a moderate positive relationship between monthly advertising expenses and monthly sales.
Tips
- Confusing R² and r: R² explains the proportion of the variance explained, while r indicates the direction and strength of a linear relationship. Remember to calculate r by taking the square root of R².
- Misinterpreting the percentage: Sometimes users forget to convert the percentage to a decimal before calculations.
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