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Chapter 13: Problem 41

The following table, reproduced from Exercise \(13.26\), gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 varieties of Kellogg's cereal. $$ \begin{array}{l|rrrrrrrrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 & 2 & 7 & 14 & 20 & 3 & 13 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 & 100 & 120 & 190 & 190 & 110 & 120 \\ \hline \end{array} $$ a. Determine the standard deviation of errors. b. Find the coefficient of determination and give a brief interpretation of it.

Short Answer

Expert verified
a. The standard deviation of errors requires step-by-step calculations starting from finding the means, slope, intercept, and using these to finally calculate the standard deviation. b. The coefficient of determination can be calculated using the formula mentioned and it explains the proportion of the variance for a dependent variable that's explained by an independent variable.

Step by step solution

01

Calculation of the variables

Firstly, calculate the mean of Sugar, mean of Calories, sum of Sugar (S), sum of Calories (C), sum of product of Sugar and Calories (SC), and the sum of squares of Sugar (SS).
02

Calculation of slope

Substitute the above obtained values in the formula of slope, \(b = \frac{N(SC)鈭�(S)(C)}{N(SS)鈭�(S)^2}\), where \(N\) is the number of observations, which is \(13\) in this case.
03

Calculation of intercept

\(\overline{Y}=b \overline{X}+a\), here \(\overline{Y}\) is the mean of Calories, \(\overline{X}\) is the mean of Sugar and \(b\) is slope from step 2. Solve this equation for \(a\), the y-intercept.
04

Calculation of values with line equation

With the equation of the line obtained from step 3, \( Y' = bX + a\), calculate the predicted values of Calories, \(Y'\).
05

Calculation of errors

Calculate the errors, which are \( E = Y - Y' \), where \(Y\) are the original Calories values and \(Y'\) are the predicted Calories values from step 4.
06

Calculation of standard deviation of errors

Find the standard deviation of errors, using the formula \(SD_{E}=\sqrt{\frac{\sum{(E)^{2}}}{n}}\), where \(n\) is the number of observations. This is the answer to part a of the exercise.
07

Calculation of coefficient of determination

Calculate coefficient of determination with the formula \(r^{2}\) = \(1 - \frac{\sum (E)^{2}}{\sum (Y - \overline{Y})^{2}}\). This is the answer to part b of the exercise.
08

Interpret coefficient of determination

A brief interpretation of this coefficient will be that it represents the proportion of the variance for a dependent variable that's explained by an independent variable.

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