Question

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

1. the equation for this least - squares regression line is (hat{y}=1.2984x + 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 th 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:

Question 5

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

Given the regression equation \(\hat{y}=1.2984x + 13.6552\), substitute \(x = 4\).

$$
LATEXBLOCK0
$$

Step2: Calculate the value of \(\hat{y}\)

$$ \hat{y}=18.8488\approx18.85 $$

The slope of a regression line \(y = mx + b\) (where \(m\) is the slope and \(x\) is the independent variable) represents the change in \(y\) for a one - unit change in \(x\). In the equation \(\hat{y}=1.2984x + 13.6552\), \(x\) is the number of toppings and \(y\) is the price of the pizza. So, for every 1 pizza topping added (\(x\) increases by 1), the price of the pizza (\(y\)) goes up by approximately \(1.30\) (the value of the slope \(m = 1.2984\approx1.30\)).

• Option A: The outlier \((1,5)\) is below the general trend of the data. Removing it would make the relationship between the number of toppings and pizza price more in line with the positive trend. The slope of the regression line (which represents the rate of increase of price with respect to toppings) would increase as the remaining data points would suggest a steeper positive relationship.
• Option B: The \(y\) - intercept is the value of \(y\) when \(x = 0\). Removing the outlier \((1,5)\) (which is an extreme low - value for \(y\) when \(x\) is small) would not necessarily decrease the \(y\) - intercept. In fact, since the overall trend is positive and the outlier is a low - value, removing it might make the \(y\) - intercept more representative of the non - outlier data (but not in a decreasing way).
• Option C: The correlation coefficient \(r\) measures the strength and direction of a linear relationship. Removing an outlier that is not in line with the general trend (the general trend is positive, and the outlier \((1,5)\) is a deviation from the positive trend) would increase the correlation (make the linear relationship stronger), not decrease it.
• Option D: The standard deviation of the residuals measures the spread of the data points around the regression line. Removing an outlier (a point with a large residual) would decrease the standard deviation of the residuals.
• Option E: The coefficient of determination \(R^{2}\) (which is \(r^{2}\)) measures the proportion of the variance in the dependent variable that is predictable from the independent variable. Since removing the outlier increases the correlation \(r\) (as explained in Option C), \(R^{2}\) would increase. But the slope also changes, and the main effect among the options is that the slope increases.

Answer:

A. $18.85

Question 6