Chapter 4, Problem 52P

Bus and subway ridership for the summer months in London, England, is believed to be tied heavily to the number of tourists visiting the city. During the past 12 years, the data on the next page have been obtained:

a) Plot these data and decide if a linear model is reasonable.

b) Develop a regression relationship.

c) What is expected ridership if 10 million tourists visit London in a year?

d) Explain the predicted ridership if there are no tourists at all.

e) What is the standard error of the estimate?

f) What is the model’s correlation coefficient and coefficient of determination?

Summary Introduction

To determine: To plot the data and decide whether the linear model is reasonable.

Introduction: Forecasting is used to predict future changes or demand patterns. It involves different approaches and varies with different periods.

Answer to Problem 52P

The graph for the given data is plotted and it can be observed that the data points are scattered around.

Explanation of Solution

Given information:

Year (Summer Months)	Number of tourist (in millions)	Ridership (in millions)
1	7	1.5
2	2	1
3	6	1.3
4	4	1.5
5	14	2.5
6	15	2.7
7	16	2.4
8	12	2
9	14	2.7
10	20	4.4
11	15	3.4
12	7	1.7

Table 1

Graphical representation:

The data to plot the graph is taken from Table 1.

Hence, the graph for the given data is plotted and it can be observed that the data points are scattered around.

Summary Introduction

To determine: A regression relationship.

Answer to Problem 52P

The linear regression equation is $\widehat{\text{y}}=0.511+\text{0}\text{.159x}$.

Explanation of Solution

Given information:

Year (Summer Months)	Number of tourist (in millions)	Ridership (in millions)
1	7	1.5
2	2	1
3	6	1.3
4	4	1.5
5	14	2.5
6	15	2.7
7	16	2.4
8	12	2
9	14	2.7
10	20	4.4
11	15	3.4
12	7	1.7

Formula of least square regression:

$\widehat{\text{y}}=\text{a}+\text{bx}$

Where,

$\begin{array}{c}\widehat{\text{y}}=\text{computed value of the variable}\\ \text{a}=\text{y-axis intercept}\\ \text{b}=\text{slope of the regression line}\\ \text{x}=\text{the independent variable}\end{array}$

$\text{b}=\frac{\sum \text{xy}-n\overline{\text{x}}\overline{\text{y}}}{\sum {\text{x}}^{\text{2}}-\text{n}{\overline{\text{x}}}^{\text{2}}}$

Where,

$\begin{array}{l}\text{b}=\text{slope of the regression line}\\ \sum =\text{ summation sign}\\ \text{x}=\text{known values of the independent variables}\\ \text{y}=\text{known values of the dependent variables}\end{array}$

$\begin{array}{l}\overline{\text{x}}=\text{average of the x - values}\\ \overline{\text{y}}=\text{average of the y - values}\\ \text{n }=\text{ number of data points}\end{array}$

Year (Summer Months)	Number of tourist (in millions) (x)	Ridership (in millions) (y)	xy	x^2	y^2
1	7	1.5	10.5	49	2.25
2	2	1	2	4	1
3	6	1.3	7.8	36	1.69
4	4	1.5	6	16	2.25
5	14	2.5	35	196	6.25
6	15	2.7	40.5	225	7.29
7	16	2.4	38.4	256	5.76
8	12	2	24	144	4
9	14	2.7	37.8	196	7.29
10	20	4.4	88	400	19.36
11	15	3.4	51	225	11.56
12	7	1.7	11.9	49	2.89
Total	132	27.1	352.9	1796	71.59

Table 2

Excel worksheet:

Substitute the values in the above formula.

Calculation of the average of x values $\overline{x}$:

$\begin{array}{c}\overline{\text{x}}=\frac{{\displaystyle \sum _{i=1}^{12}\text{x}}}{n}\\ =\frac{132}{12}\\ =11\end{array}$

The average of x values is obtained by dividing the summation of x values with the number of periods n=12, the value of $\overline{x}$= 11.

Calculation of the average of y values $\overline{y}$:

$\begin{array}{c}\overline{\text{y}}=\frac{{\displaystyle \sum _{i=1}^{12}y}}{n}\\ =\frac{27.1}{12}\\ =2.26\end{array}$

The average of y values is obtained by dividing the summation of sales with the number of periods n=12. The value of $\overline{y}$= 2.26.

Calculation ofthe slope of regression line‘b’:

$\begin{array}{c}\text{b}=\frac{\sum \text{xy}-n\overline{\text{x}}\overline{\text{y}}}{\sum {\text{x}}^{\text{2}}-n{\overline{\text{x}}}^{\text{2}}}\\ =\frac{352.9-\left(12\times 11\times 2.26\right)}{1796-\left(12\times {11}^2\right)}\\ =\frac{54.6}{344}\\ =0.159\end{array}$

The summation of the product of sales (y) with x values is ∑xy = 352.9, the product of number of period (n), the average of x values and the average of y values is obtained; $n\overline{\text{x}}\overline{\text{y}}$=298.3. The difference between 352.9and 298.3 is 54.6.

The summation of the square of x values, 1796, is subtracted from the product of the number of periods, 10with the average of x values, 11. The resultant value is 344. The slope of the regression line is obtained by dividing 1796 with 344. The value of ‘b’ is 0.159.

Calculation of the y-axis intercept ‘a’:

$\begin{array}{c}\text{a}=\overline{\text{y}}-\text{b}\overline{\text{x}}\\ =2.26-\left(0.159\times 11\right)\\ =0.511\end{array}$

The y-axis intercept is obtained by the difference between the average of y values and values obtained by the product of the slope of regression line with the average of x values. The resultant value of ‘a’ is 0.511.

Least Square Regression forecasting equation:

$\begin{array}{c}\widehat{\text{y}}=\text{a}+\text{bx}\\ =0.511+\text{0}\text{.159x}\end{array}$ (1)

Substitute the slope of regression line and they axis intercept in the regression equation which gives the liner regression equation for the data.

Hence, the linear regression equation is $\widehat{\text{y}}=0.511+\text{0}\text{.159x}$.

Summary Introduction

To determine: The expected ridership when 10 million tourists visit in a year.

Answer to Problem 52P

There is a 2.101 million ridership when 10 million tourists visit in a year.

Explanation of Solution

Given information:

Year (Summer Months)	Number of tourist (in millions)	Ridership (in millions)
1	7	1.5
2	2	1
3	6	1.3
4	4	1.5
5	14	2.5
6	15	2.7
7	16	2.4
8	12	2
9	14	2.7
10	20	4.4
11	15	3.4
12	7	1.7

Formula of least square regression:

$\widehat{\text{y}}=\text{a}+\text{bx}$

Where,

$\begin{array}{c}\widehat{\text{y}}=\text{computed value of the variable}\\ \text{a}=\text{y-axis intercept}\\ \text{b}=\text{slope of the regression line}\\ \text{x}=\text{the independent variable}\end{array}$

$\text{b}=\frac{\sum \text{xy}-n\overline{\text{x}}\overline{\text{y}}}{\sum {\text{x}}^{\text{2}}-\text{n}{\overline{\text{x}}}^{\text{2}}}$

Where,

$\begin{array}{l}\text{b}=\text{slope of the regression line}\\ \sum =\text{ summation sign}\\ \text{x}=\text{known values of the independent variables}\\ \text{y}=\text{known values of the dependent variables}\end{array}$

$\begin{array}{l}\overline{\text{x}}=\text{average of the x - values}\\ \overline{\text{y}}=\text{average of the y - values}\\ \text{n }=\text{ number of data points}\end{array}$

Calculation of number of ridership when 10 million touristsvisit in a year:

$\begin{array}{c}\widehat{\text{y}}=0.511+\text{0}\text{.159x}\\ \text{=0}\text{.511+}\left(0.159\times 10\right)\\ =2.101\text{million}\text{persons}\end{array}$

Equation (1) provides the linear regression equation for the data and substitutes the number of tourists visiting in the regression equation. Substituting 10 million in the equation, the resultant value is found to be 2.101 million ridership.

Hence, there are 2.101 million ridership when 10 million touristsvisit in a year.

Summary Introduction

To determine: The expected ridership when no tourists visit in a year.

Answer to Problem 52P

There is a 511,000 ridership when no touristsvisit in a year.

Explanation of Solution

Given information:

Year (Summer Months)	Number of tourist (in millions)	Ridership (in millions)
1	7	1.5
2	2	1
3	6	1.3
4	4	1.5
5	14	2.5
6	15	2.7
7	16	2.4
8	12	2
9	14	2.7
10	20	4.4
11	15	3.4
12	7	1.7

Formula of least square regression:

$\widehat{\text{y}}=\text{a}+\text{bx}$

Where,

$\begin{array}{c}\widehat{\text{y}}=\text{computed value of the variable}\\ \text{a}=\text{y-axis intercept}\\ \text{b}=\text{slope of the regression line}\\ \text{x}=\text{the independent variable}\end{array}$

$\text{b}=\frac{\sum \text{xy}-n\overline{\text{x}}\overline{\text{y}}}{\sum {\text{x}}^{\text{2}}-\text{n}{\overline{\text{x}}}^{\text{2}}}$

Where,

$\begin{array}{l}\text{b}=\text{slope of the regression line}\\ \sum =\text{ summation sign}\\ \text{x}=\text{known values of the independent variables}\\ \text{y}=\text{known values of the dependent variables}\end{array}$

$\begin{array}{l}\overline{\text{x}}=\text{average of the x - values}\\ \overline{\text{y}}=\text{average of the y - values}\\ \text{n }=\text{ number of data points}\end{array}$

Calculation of the number of ridership when notouristsvisit in a year:

$\begin{array}{c}\widehat{\text{y}}=0.511+\text{0}\text{.159x}\\ \text{=0}\text{.511+}\left(0.159\times 0\right)\\ =0.511\text{million}\text{persons}\\ \text{=511,000}\text{persons}\end{array}$

Equation (1) provides the linear regression equation for the data and substitutes the number of tourists visiting in the regression equation. Substituting 0 in the equation, the resultant value is 0.511 million ridership.

Hence, there is a 511,000 ridership when notouristsvisit in a year.

Summary Introduction

To determine: The standard error of estimate.

Answer to Problem 52P

The standard error of estimate is0.4037.

Explanation of Solution

Given information:

Year (Summer Months)	Number of tourists(in millions)	Ridership (in millions)
1	7	1.5
2	2	1
3	6	1.3
4	4	1.5
5	14	2.5
6	15	2.7
7	16	2.4
8	12	2
9	14	2.7
10	20	4.4
11	15	3.4
12	7	1.7

Formula to compute the standard error of estimate:

$\begin{array}{c}{\text{S}}_{\text{yx}}=\sqrt{\frac{\sum {(\text{y}-{\text{y}}_{\text{c}})}^2}{n-2}}\\ \text{y}=\text{value}\text{of}\text{each}\text{data}\text{point}\\ {\text{y}}_{\text{c}}=\text{Computed}\text{value}\text{of}\text{the}\text{dependent}\text{variable}\text{from}\text{the}\text{regression}\text{equation}\\ n=\text{number}\text{of}\text{data}\text{points}\end{array}$

${\text{s}}_{\text{yx}}=\sqrt{\frac{\sum {\text{y}}^2-\text{a}\sum \text{y}-\text{b}\sum \text{xy}}{n-2}}$

Calculation of standard error of estimate:

$\begin{array}{c}{\text{S}}_{\text{yx}}=\sqrt{\frac{\sum {\text{y}}^2-\text{a}\sum \text{y}-\text{b}\sum \text{xy}}{n-2}}\\ =\sqrt{\frac{71.59-\left(0.511\times 27.1\right)-\left(0.159\times 352.9\right)}{12-2}}\\ =\sqrt{0.163}\\ =0.4037\end{array}$

The values to be substituted in the standard error of estimate formula are given inTable 2. Substitute the values from the table in the formula. This results in a standard error of estimate of 0.4037.

Hence, the standard error of estimate is 0.4037.

Summary Introduction

To determine: The coefficient of correlation (r) and coefficient of determination (r²).

Answer to Problem 52P

The coefficient of correlation (r) and coefficient of determination (r²) are 717.41 & 0.840, respectively.

Explanation of Solution

Given information:

Year (Summer Months)	Number of tourists(in millions)	Ridership (in millions)
1	7	1.5
2	2	1
3	6	1.3
4	4	1.5
5	14	2.5
6	15	2.7
7	16	2.4
8	12	2
9	14	2.7
10	20	4.4
11	15	3.4
12	7	1.7

Formula to calculate the correlation coefficient:

$r=\frac{n\sum \text{xy}-\sum \text{x}\sum \text{y}}{\sqrt{\left(n\sum {\text{x}}^2-{\left(\sum \text{x}\right)}^2{\left(n\sum {\text{y}}^2-\left(\sum \text{y}\right)\right)}^2\right)}}$

Calculation of the correlation coefficient (r):

Table (2) provides the values to calculate the correlation coefficient (r).

$\begin{array}{c}r=\frac{n\sum \text{xy}-\sum \text{x}\sum \text{y}}{\sqrt{\left(n\sum {\text{x}}^2-{\left(\sum \text{x}\right)}^2{\left(n\sum {\text{y}}^2-\left(\sum \text{y}\right)\right)}^2\right)}}\\ =\frac{\left(12\times 352.9\right)-\left(132\times 27.1\right)}{\sqrt{\left(\left(12\times 1796\right)-{132}^2\right)\left(\left(12\times 71.59\right)-{\left(27.1\right)}^2\right)}}\\ =\frac{657.6}{2934.31}\\ =717.41\end{array}$

Calculation of the correlation of determination (r²):

$\begin{array}{c}{\text{r}}^2=\text{r}\times \text{r}\\ =0.917\times 0.917\\ =0.840\end{array}$

Hence, the coefficient of correlation and coefficient of determination are 717.41 and 0.840, respectively.