Chapter 2, Problem 54AP

a.

Summary Introduction

To calculate: The dollar sales of ice cream in two years, based on the regression equation treating ice cream sales as the dependent variable and time as the independent variable

Introduction: Regression analysis is a statistical method for estimating the interactions between variables. In data forecasting, companies can learn trends using regression analysis. It enables data based predictions to be made.

Answer to Problem 54AP

The dollar sales of ice cream in two years:

  $30\times 24\times 4244.92=\boxed{\$30,563,42.4}$

Explanation of Solution

Month	Park Attendies $\left(X_i\right)$	Ice cream sales $\left(Y_i\right)$	$X_i{Y}_i$	$X_i{}^2$
1	880	325	286,000	77,440
2	976	335	326,926	95,576
3	440	172	75,680	193,600
4	1,823	645	1,175,835	3,323,329
5	1,885	770	1,451,450	3,553,225
6	2,436	950	2,314,200	5,934,096
Total = 21	8440	3197	5,630,125	13,177,266

  $\begin{array}{l}{S}_{xx}=\frac{n^2\left(n+1\right)\left(2n+1\right)}{6}-\frac{n^2{\left(n+1\right)}^2}{4}\mathrm{......}\left(1\right)\\ S_{xy}=n{\displaystyle \sum _{i=1}^{n}i{D}_i}-\frac{n\left(n+1\right)}{2}{\displaystyle \sum _{i=1}^n{D}_i}\end{array}$

Substituting the values in equation (1)

  $\begin{array}{c}{S}_{xx}=\frac{n^2\left(n+1\right)\left(2n+1\right)}{6}-\frac{n^2{\left(n+1\right)}^2}{4}\\ =\frac{6^2\left(6+1\right)\left(2\times 6+1\right)}{6}-\frac{6^2{\left(6+1\right)}^{2}4}{4}\\ =105\end{array}$

  $\begin{array}{c}{S}_{xy}=n{\displaystyle \sum _{i=1}^{n}i{D}_i}-\frac{n\left(n+1\right)}{2}{\displaystyle \sum _{i=1}^n{D}_i}\\ =6{\displaystyle \sum _{i=1}^{n}i{D}_i}-\frac{6\left(6+1\right)}{2}{\displaystyle \sum _{i=1}^n{D}_i}\\ =14,709\end{array}$

Calculating general regression using the following equations:

  $b=\frac{S_{xy}}{S_{xx}}\mathrm{.....}\left(2\right)$

  $a=\overline{y}-b\overline{x}$

Now,

  $\begin{array}{l}b=\frac{S_{xy}}{S_{xx}}=\frac{14709}{105}=\boxed{140.08}\\ a=\overline{y}-b\overline{x}\\ \overline{y}=\frac{1}{n}{\displaystyle \sum y_i}=\frac{3197}{6}=532.83\\ \overline{x}=\frac{1}{n}{\displaystyle \sum x_i=\frac{21}{6}=3.5}\\ a=532.83-\left(140.08\right)\left(3.5\right)=\boxed{42.52}\end{array}$

Calculating the regression equation below:

  $\begin{array}{l}{Y}_t=a+bX\\ Y_{30}=42.52+\left(30\times 140.08\right)\\ \boxed{Y_{30}=4244.92}\end{array}$

The forecast predicted does not seem confident enough to assume as the trend observed over the first six months might be unlikely for the next six months of the year.

b.

Summary Introduction

To calculate: A regression equation treating ice cream sales as the dependent variable and part attendees as the independent variable

Introduction: Regression analysis is a statistical method for estimating the interactions between variables. In data forecasting, companies can learn trends using regression analysis. It enables data based predictions to be made.

Answer to Problem 54AP

  $\begin{array}{l}Y=a+bX\\ \boxed{y=-24.207+0.396X}\end{array}$

Explanation of Solution

Month	Park Attendies $\left(X_i\right)$	Ice cream sales $\left(Y_i\right)$	$X_i{Y}_i$	$X_i{}^2$
1	880	325	286,000	77,440
2	976	335	326,926	95,576
3	440	172	75,680	193,600
4	1,823	645	1,175,835	3,323,329
5	1,885	770	1,451,450	3,553,225
6	2,436	950	2,314,200	5,934,096
Total = 21	8440	3197	5,630,125	13,177,266

  $\begin{array}{l}{S}_{xx}=\frac{n^2\left(n+1\right)\left(2n+1\right)}{6}-\frac{n^2{\left(n+1\right)}^2}{4}\mathrm{......}\left(3\right)\\ S_{xy}=n{\displaystyle \sum _{i=1}^{n}i{D}_i}-\frac{n\left(n+1\right)}{2}{\displaystyle \sum _{i=1}^n{D}_i}\end{array}$

Substituting the values in equation (3)

  $\begin{array}{c}{S}_{xy}=n{\displaystyle \sum _{i=1}^{n}i{D}_i}-\frac{n\left(n+1\right)}{2}{\displaystyle \sum _{i=1}^n{D}_i}\\ =6{\displaystyle \sum _{i=1}^{n}i{D}_i}-\frac{6\left(6+1\right)}{2}{\displaystyle \sum _{i=1}^n{D}_i}\\ =6,798,070\end{array}$

  $\begin{array}{l}{S}_{xx}=\frac{n^2\left(n+1\right)\left(2n+1\right)}{6}-\frac{n^2{\left(n+1\right)}^2}{4}\mathrm{......}\left(3\right)\\ S_{xx}=17,153,1576\end{array}$

Calculating general regression using the following equations:

Now,

  $\begin{array}{l}b=\frac{S_{xy}}{S_{xx}}=\frac{6,798,070}{17,1531576}=\boxed{0.396}\\ a=\overline{y}-b\overline{x}\\ \overline{y}=\frac{1}{n}{\displaystyle \sum y_i}=\frac{3197}{6}=532.83\\ \overline{x}=\frac{1}{n}{\displaystyle \sum x_i=\frac{8440}{6}=1406.66}\\ a=532.83-\left(0.396\right)\left(1406.66\right)=\boxed{-24.207}\end{array}$

c.

Summary Introduction

To calculate: Ice cream sales for months 12 through 18, based on the curve and the regression equation determined in part (b)

Introduction: Regression analysis is a statistical method for estimating the interactions between variables. In data forecasting, companies can learn trends using regression analysis. It enables data based predictions to be made.

Answer to Problem 54AP

  $\begin{array}{l}{Y}_{12}=1995.39\\ Y_{13}=2094.39\\ Y_{14}=2193.39\\ Y_{15}=2272.59\\ Y_{16}=2312.19\\ Y_{17}=2331.99\\ Y_{18}=2343.87\end{array}$

Explanation of Solution

The following logistic curve can be considered for obtaining values:

The number of attendies observed from the logistic curve from months 12 to 18 until peak 3000 is shown below:

MONTH	ATTENDIES	PREDICTED ICE CREAM SALES
12	5100	1995.39
13	5350	2094.39
14	5600	2193.39
15	5800	2272.59
16	5900	2312.19
17	5950	2331.99
18	5980	2343.87

Using the regression equation $y=-24.207+0.396X$ , the calculations of the forecast sales are shown below:

  $Y_t=a+bX$

  $\begin{array}{l}{Y}_{12}=-24.207+\left(0.396\times 5100\right)\\ \boxed{Y_{12}=1995.39}\end{array}$

  $\begin{array}{l}{Y}_{13}=-24.207+\left(0.396\times 5350\right)\\ \boxed{Y_{13}=2094.39}\end{array}$

  $\begin{array}{l}{Y}_{14}=-24.207+\left(0.396\times 5600\right)\\ \boxed{Y_{14}=2193.39}\end{array}$

  $\begin{array}{l}{Y}_{15}=-24.207+\left(0.396\times 5800\right)\\ \boxed{Y_{15}=2272.59}\end{array}$

  $\begin{array}{l}{Y}_{16}=-24.207+\left(0.396\times 5900\right)\\ \boxed{Y_{16}=2312.19}\end{array}$

  $\begin{array}{l}{Y}_{17}=-24.207+\left(0.396\times 5950\right)\\ \boxed{Y_{17}=2331.19}\end{array}$

  $\begin{array}{l}{Y}_{18}=-24.207+\left(0.396\times 5980\right)\\ \boxed{Y_{18}=2343.87}\end{array}$