Question

valentino recorded the number of lemonades ordered at his restaurant and the corresponding average monthly temperature for the past 15 months. he displayed the information in a scatter plot and used a graphing calculator program to find the least squares regression line to fit the data. valentino wants to use the least squares regression line to predict the number of lemonades that will be ordered when the average monthly temperature is 65°f. which statement is true? exactly 122 lemonades will be ordered when the average monthly temperature is 65°f. about 118 lemonades will be ordered for an average monthly temperature of 65°f. valentino cannot use this line to make predictions because it does not fit the data well. valentino cannot use this line to make predictions because there is no data for an average monthly temperature of 65°f.

Explanation:

Step1: Understand the nature of regression line

A regression line gives an estimate, not an exact value. So the statement "Exactly 122 lemonades will be ordered when the average monthly temperature is \(65^{\circ}F\)" is wrong.

Step2: Check the fit of the regression line

The scatter - plot shows a clear positive linear trend, so the regression line fits the data well. Thus, the statement "Valentino cannot use this line to make predictions because it does not fit the data well" is wrong.

Step3: Consider the concept of prediction using regression

Regression can be used for prediction within the range of the data (interpolation). \(65^{\circ}F\) is within the range of \(x\) - values (temperature values) in the scatter - plot. So the statement "Valentino cannot use this line to make predictions because there is no data for an average monthly temperature of \(65^{\circ}F\)" is wrong.

Step4: Estimate using the regression line

By looking at the general trend of the scatter - plot and the regression line (if we assume a linear relationship \(y = mx + b\) and estimate based on the position of \(x = 65\) on the \(x\) - axis and the corresponding \(y\) - value on the regression line), we can see that about 118 lemonades is a reasonable estimate.

Answer:

About 118 lemonades will be ordered for an average monthly temperature of \(65^{\circ}F\).