Textbook Question
Why don’t we conduct inference on the linear correlation coefficient?
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11. Correlation
Correlation Coefficient
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A marketing researcher analyzed advertising budget vs. monthly sales revenue for small retail stores and found that typically the stores that spent more on advertising saw higher sales revenues. However, the relationship wasn't perfect - some stores advertised more but saw fewer sales due to poor location, customer preferences, or bad timing. Which of the following is the most likely value for the correlation coefficient between advertising budget and sales revenue?
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Verified step by step guidance1
Step 1: Understand the concept of the correlation coefficient (r). The correlation coefficient measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 indicate a strong positive relationship, values close to -1 indicate a strong negative relationship, and values near 0 indicate little to no linear relationship.
Step 2: Analyze the context of the problem. The problem states that stores with higher advertising budgets generally see higher sales revenues, but the relationship is not perfect due to factors like location, customer preferences, or timing. This suggests a positive but not perfect correlation.
Step 3: Eliminate options that do not fit the context. A correlation of r = -0.12 or r = -0.86 would indicate a negative relationship, which contradicts the observation that higher advertising budgets are generally associated with higher sales revenues. These options can be ruled out.
Step 4: Compare the remaining options. A correlation of r = 0.96 indicates a very strong positive relationship, which would imply that advertising budget almost perfectly predicts sales revenue. However, the problem mentions that the relationship is not perfect, so this value is unlikely.
Step 5: Conclude that the most likely value for the correlation coefficient is r = 0.59, as it represents a moderate positive relationship, consistent with the described scenario where higher advertising budgets are generally associated with higher sales revenues, but other factors also play a role.
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