Question

the developer of a new plant fertilizer gathered data about the quantity of fertilizer applied to a lemon tree and the number of lemons it produced. he used a graphing tool to organize the data in a scatter plot and find the line of best fit. he found that the relationship between grams of fertilizer, x, and the number of lemons, y, is modeled by the equation y = 0.316x + 12.214, and the correlation coefficient for the data is 0.948. could this line of best fit be used to make reliable predictions? a. no, because the slope of the line is closer to 0 than to 1, which indicates a weak association between the variables. b. no, because the correlation coefficient is closer to 1 than to 0, which indicates a weak association between the variables. c. yes, because the slope of the line is closer to 0 than to 1, which indicates a strong association between the variables. d. yes, because the correlation coefficient is closer to 1 than to 0, which indicates a strong association between the variables.

Explanation:

Step1: Recall correlation - coefficient concept

The correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables.

Step2: Analyze the value of given correlation - coefficient

The given correlation coefficient \(r = 0.948\). Since \(|r|\) is close to 1 (where \(|r|\in[0,1]\)), it indicates a strong linear relationship between the variables. A line of best - fit can be used to make reliable predictions when there is a strong linear relationship. The slope of the line does not determine the strength of the association; the correlation coefficient does.

Answer:

D. Yes, because the correlation coefficient is closer to 1 than to 0, which indicates a strong association between the variables.