Question

select the correct answer.

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?

no, because the slope of the line is closer to 0 than to 1, which indicates a weak association between the variables.
no, because the correlation coefficient is closer to 1 than to 0, which indicates a weak association between the variables.
yes, because the slope of the line is closer to 0 than to 1, which indicates a strong association between the variables.
yes, because the correlation coefficient is closer to 1 than to 0, which indicates a strong association between the variables.

Explanation:

Analyze the correlation coefficient

The correlation coefficient \(r\) is given as \(0.948\). This value measures the strength and direction of the linear relationship between the two variables.

Evaluate the strength of the association

Since \(r = 0.948\) is very close to \(1\), it indicates a very strong positive linear association between the grams of fertilizer applied and the number of lemons produced.

Determine the reliability of predictions

A strong association (where the correlation coefficient is close to \(1\) or \(-1\)) means the line of best fit is a highly reliable model for making predictions. Therefore, the line of best fit can be used to make reliable predictions because the correlation coefficient is closer to \(1\) than to \(0\), indicating a strong association.

Answer:

• No, because the slope of the line is closer to 0 than to 1, which indicates a weak association between the variables.
• No, because the correlation coefficient is closer to 1 than to 0, which indicates a weak association between the variables.
• Yes, because the slope of the line is closer to 0 than to 1, which indicates a strong association between the variables.
• Yes, because the correlation coefficient is closer to 1 than to 0, which indicates a strong association between the variables. (Correct answer)