Chapter 4, Problem 28P

A

Summary Introduction

Interpretation:Based on the expected average air temperature, estimate the next month sales using the linear regression model.

Concept Introduction: Using regression, we will be able to define relationship between any two variables, denoting the cause and effect. The method can also be used to forecast the future depending on the past performances.

Answer to Problem 28P

After necessary calculations, we can observe that the slope value is -1.626.

Explanation of Solution

Given Information:

The number of months is 42, n.

Multiplied value of average temperature and average sales is 873,931,∑xy.

Sum of squared monthly temperature is 105,080,∑x².

Sum of air temperature reading is 2,058,∑x.

Sum of sales is 17,976,∑y.

  $\begin{array}{l}Slope(b)=\frac{n{\displaystyle \sum xy-{\displaystyle \sum x{\displaystyle \sum y}}}}{n{{\displaystyle \sum x^2-\left({\displaystyle \sum x}\right)}}^2}\\ =\frac{42(873,931)-(2058)(17976)}{42(105080)-{(2058)}^2}\\ =\frac{36705102-36994608}{4413360-4235364}\\ =\frac{-289506}{177996}\\ =-1.626\end{array}$

Now, we will derive intercept,

  $\begin{array}{l}Intercept(a)=\frac{{\displaystyle \sum y-b{\displaystyle \sum x}}}{n}\\ =\frac{17,976-(-1.626\times 2058)}{42}\\ =\frac{17976+3346.308}{42}\\ =\frac{21322.308}{42}\\ =507.64\end{array}$

Thus, the value of intercept is 507.64.

Now, substitute these values in the below equation,

  $\begin{array}{l}y=a+bx\\ y=507.67+(-1.626\times 35)\\ y=507.67-56.91\\ y=450.8\end{array}$

This means, the approximate value is 451 rolls insulation.

B

Summary Introduction

Interpretation:Determine the percent of variation in the sales in the past 42 months, by observing the linear relationship to the average temperature.

Concept Introduction: Using regression, we will be able to define relationship between any two variables, denoting the cause and effect. The method can also be used to forecast the future depending on the past performances.

Answer to Problem 28P

After necessary calculations, we can observe that the percent of variation is 68.56%.

Explanation of Solution

Given Information:

Number of months, n is 42 months.

Multiplied value of average temperature and average sales,∑xy is 873,931.

Sum of squared monthly temperature,∑x² is 105,080.

Sum of air temperature reading,∑x is 2058.

Sum of sales,∑y is 17976.

Sum of squared monthly sales,∑y² is 7710080.

49 degrees of average air temperature, x-.

428 rolls of average sales, y-.

First, let us find out correlation coefficient,

  $\begin{array}{l}\text{Correlation Coefficient=}\frac{{\displaystyle \sum xy-n\overline{x}\overline{y}}}{\sqrt{[({\displaystyle \sum x^2-n{\overline{x}}^2})]\times [({\displaystyle \sum y^2-n{\overline{y}}^2})]}}\\ =\frac{873,931-(42\times 49\times 428)}{\sqrt{[105,080-(42\times {49}^2)\times 7710080-(42\times {428}^2)}}\\ =\frac{-6893}{8324.65}\\ =-0.828\end{array}$

Now, we shall determine the percentage of variation.

  $\begin{array}{l}\text{Percent of Variation=Correlation Coefficient}\times \text{Correlation Coefficient}\times \text{100\%}\\ \text{=(-0}\text{.828)}\times \text{(-0}\text{.828)}\times 100\%\\ =68.56\%\end{array}$

C

Summary Introduction

Interpretation: Determine the correlation coefficient between sales and average air temperature, and also briefly put out the relationship between those two.

Concept Introduction: Using regression, we will be able to define relationship between any two variables, denoting the cause and effect. The method can also be used to forecast the future depending on the past performances.

Answer to Problem 28P

The correlation coefficient is -0.828, and this indicates that the sales will increase with a decrease in the air temperature.

Explanation of Solution

Given Information:

Number of months, n is 42 months.

Multiplied value of average temperature and average sales,∑xy is 873,931.

Sum of squared monthly temperature,∑x² is 105,080.

Sum of air temperature reading,∑x is 2058.

Sum of sales,∑y is 17976.

Sum of squared monthly sales,∑y² is 7710080.

49 degrees of average air temperature, x-.

428 rolls of average sales, y-.

  $\begin{array}{l}\text{Correlation Coefficient=}\frac{{\displaystyle \sum xy-n\overline{x}\overline{y}}}{\sqrt{[({\displaystyle \sum x^2-n{\overline{x}}^2})]\times [({\displaystyle \sum y^2-n{\overline{y}}^2})]}}\\ =\frac{873,931-(42\times 49\times 428)}{\sqrt{[105,080-(42\times {49}^2)\times 7710080-(42\times {428}^2)}}\\ =\frac{-6893}{8324.65}\\ =-0.828\end{array}$

Thus, we can see that there will be an increase in the sales with a decrease in the air temperature.