Question

a. Is a line an appropriate model to use for these data? Explain how you know.

b. Find the correlation.

c. What is the equation of the least-squares regression line? Define any variables that you

use.

d. Interpret the values of s and r2.

More mess? When Mentos are dropped into a newly opened bottle of Diet

Coke, carbon dioxide is released from the Diet Coke very rapidly, causing the Diet Coke to be expelled from the bottle. To see if using more Mentos causes more Diet Coke to be expelled, Brittany and Allie used twenty-four 2-cup bottles of Diet Coke and randomly assigned each bottle to receive either 2, 3, 4, or 5 Mentos. After waiting for the fizzing to stop, they measured the amount expelled (in cups) by subtracting the amount remaining from the original amount in the bottle.29 Here is some computer output from a regression of y = amount expelled on x = number of Mentos:

a. Is a line an appropriate model to use for these data? Explain how you know.

b. Find the correlation.

c. What is the equation of the least-squares regression line? Define any variables that you use.

d. Interpret the values of s and r2.

Short answer

Part (a) Yes, a line appears to be appropriate for these data.

Part (b)$r=0.7760$

Part (c) $\stackrel{⏜}{y}=1.0021+0.0708x$

Part (d) The least square regression line utilizing the number of Mentos as an explanatory variable can explain $60.21$ percent of the variation in the amount ejected.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

The question specifies the relationship between the number of Mentos consumed and the amount of cocaine ejected. For the same, a scatterplot and a residual plot are provided. Because there is no substantial curvature in the scatterplot and no strong curvature in the residual plot, it appears that a line is appropriate for these data.

Part (b) Step 1: Explanation

The relationship between the number of Mentos and the amount of coke expelled is given in the question. The scatterplot and the residual plot for the same are given. Thus, the coefficient of determination is given in the computer output after “R-Sq” as:

$$\begin{gathered} r^2=60.21\% \\ =0.6021 \end{gathered}$$

The positive or negative square root of the coefficient of determination r yields the linear correlation coefficient r. As a result, we can see that the scatterplot pattern slopes upwards, indicating a positive relationship between the variables and hence a positive linear correlation coefficient $r$This indicates that,

$$\begin{gathered} r=+\sqrt{r^2} \\ =+\sqrt{0.6021} \\ =0.7760 \end{gathered}$$

Part (c) Step 1: Explanation

The question specifies the relationship between the number of Mentos consumed and the amount of cocaine ejected. For the same, a scatterplot and a residual plot are provided. The least-square regression line's general equation is:

$\stackrel{⏜}{y}=b_0+b_{1}x$

As a result, the estimate of the constant b0 is given in the computer output's row "Constant" and column "Coef" as:

$b_0=1.0021$

In the computer output, the estimate of the slope b1 is presented in the row "Mentos" and the column "Coef" as:

$b_1=0.0708$

Thus, putting the values in the general equation we will get,

$\stackrel{⏜}{y}=b_0+b_{1}x$

$\Rightarrow \stackrel{⏜}{y}=1.0021+0.0708x$

Part (d) Step 1: Explanation

The question specifies the relationship between the number of Mentos consumed and the amount of cocaine ejected. For the same, a scatterplot and a residual plot are provided. As a result, after $"S="$in the computer output, the standard error of the estimations is given as:

$s=0.6724$

The standard error of the estimations, as we all know, is the average error of forecasts, and thus the average difference between actual and predicted values. As a result, the predicted amount evacuated using the equation of the least square regression line was 0.6724 cups less than the actual amount expelled on average.

In the computer output, the coefficient of determination is now given after $"R-Sq"$as:

$$\begin{gathered} r^2=60.21\% \\ =0.6021 \end{gathered}$$

The coefficient of determination, as we all know, represents the proportion of variance in the answers y variable that can be explained by a least square regression model using the explanatory $x$ variable. As a result, the least square regression line utilizing the number of Mentos as an explanatory variable can explain $60.21$ percent of the variation in the amount ejected.