Question

use the following scatterplot for questions 5 - 7. in your area, you are curious if the number of toppings on your pizza follows a linear relationship with the price of a large pizza. you collect data from the piz in your town and compile them in the following scatterplot. the least - squares regression line has been the plot.

1. the equation for this least - squares regression line is \\( \hat { y } = 1.2984 x + 13.6552 \\). what is the predicted price of a large pizza if i order 4 toppings?

(a) \\( \\$ 18.85 \\)
(b) \\( \\$ 16.50 \\)
(c) \\( \\$ 6.52 \\)
(d) \\( \\$ 20.15 \\)
(e) \\( \\$ 15.50 \\)

1. what is the correct interpretation of the slope of the least - squares regression line?

(a) for every 1 pizza topping added, the price of the pizza will go up by \\( \\$ 13.66 \\).
(b) for every 1 pizza topping added, the price of the pizza will go up by \\( \\$ 1.30 \\).
(c) for every \\( \\$ 1 \\) increase in pizza price, the number of pizza toppings increases by approximately 13.66
(d) for every \\( \\$ 1 \\) increase in pizza price, the number of pizza toppings increases by approximately 1.3
(e) none of the above statements gives the correct interpretation of slope.

1. in the scatterplot above, there is a clear outlier (little caesars!) at the point \\( ( 1,5 ) \\). removing this from the data would have which of the following effects?

(a) the slope would increase
(b) the \\( y \\) - intercept would decrease
(c) the correlation would decrease
(d) the standard deviation of the residuals would increase
(e) the coefficient of determination would increase

Explanation:

Step1: Substitute \(x = 4\) into the regression equation

Given the regression equation \(\hat{y}=1.2984x + 13.6552\), when \(x = 4\), we have \(\hat{y}=1.2984\times4+13.6552\).
First, calculate \(1.2984\times4\): \(1.2984\times4=(1 + 0.2984)\times4=4+1.1936 = 5.1936\).
Then add \(13.6552\): \(5.1936+13.6552=18.8488\approx18.85\).

Step2: Interpret the slope of the regression line

The general form of a simple - linear regression equation is \(\hat{y}=b_0 + b_1x\), where \(b_1\) is the slope. In the equation \(\hat{y}=1.2984x + 13.6552\), the slope \(b_1 = 1.2984\approx1.30\). The slope represents the change in \(\hat{y}\) (dependent variable, here it is the price of pizza) for a one - unit change in \(x\) (independent variable, here it is the number of toppings). So for every 1 pizza topping added, the price of the pizza will go up by approximately \(\$1.30\).

Step3: Analyze the effect of removing an outlier

• Slope: The outlier \((1,5)\) is below the general trend of the data. Removing it will make the regression line better fit the remaining data points which have a positive trend. The slope will increase slightly as the remaining data has a steeper positive relationship.
• Y - intercept: Since the outlier is below the trend, removing it will make the line shift upwards (because the line is now fitting data that is generally higher for a given \(x\) value), so the \(y\) - intercept will increase.
• Correlation: The outlier weakens the positive correlation. Removing it will increase the correlation coefficient (since the remaining data has a stronger positive linear relationship).
• Standard deviation of residuals: Residuals are \(y-\hat{y}\). The outlier has a large residual. Removing it will decrease the standard deviation of residuals.
• Coefficient of determination (\(R^{2}\)): \(R^{2}\) is related to the correlation coefficient \(r\) (\(R^{2}=r^{2}\)). Since removing the outlier increases \(r\), it will increase \(R^{2}\).

Answer:

1. A. \(\$18.85\)
2. B. For every 1 pizza topping added, the price of the pizza will go up by \(\$1.30\)
3. E. the coefficient of determination would increase