Chapter 16.2, Problem 4E

(a)

To determine

Draw the scatter diagram that shows the relationship between television and overweight.

Explanation of Solution

Figure 1 shows the scatter diagram of television and overweight.

In the above figure, the vertical axis measures overweight and the horizontal axis measures television. The position of the scatters shows a positive relationship between overweight and television.

Economics Concept Introduction

Scatter diagram: The scatter diagram shows the value of two variables plotted along two axes. Each scatter represents the relationship between the variables.

(b)

To determine

Calculate the least squares line and interpret the slope of the line.

Explanation of Solution

The least squares regression line is shown below:

$\stackrel{\wedge }{\text{y}}={\text{b}}_0+{\text{b}}_1\text{x}$

Where b₀ is the y-intercept and b₁ is the slope of the regression line.

The least squares regression line is calculated as follows:

Table 1

	Xi	Yi	Xi2	Yi2	XiYI
1	42	18	1,764	324	756
2	34	6	1,156	36	204
3	25	0	625	0	0
4	35	–1	1,225	1	–35
5	37	13	1,369	169	481
6	38	14	1,444	196	532
7	31	7	961	49	217
8	33	7	1,089	49	231
9	19	–9	361	81	–171
10	29	8	841	64	232
11	38	8	1,444	64	304
12	28	5	784	25	140
13	29	3	841	9	87
14	36	14	1,296	196	504
15	18	–7	324	49	–126
Total	472	86	15,524	1,312	3,356

These table values are obtained using a spreadsheet.

$\begin{array}{c}{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}}=472,{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{y}}_{\text{i}}}=86\\ {\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}^{\text{2}}}=15,524,{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{y}}_{\text{i}}^{\text{2}}}=1,312\\ {\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}{\text{y}}_{\text{i}}}=3,356\\ {\text{s}}_{\text{xy}}=\frac{\text{1}}{\text{n}-\text{1}}\left[{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}{\text{y}}_{\text{i}}}-\frac{{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}}{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{y}}_{\text{i}}}}{\text{n}}\right]\\ =\frac{1}{15-1}\left[3,356-\frac{(472)(86}{15}\right]=46.42\\ {\text{s}}_{\text{x}}^{\text{2}}=\frac{\text{1}}{\text{n}-\text{1}}\left[{\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}^{\text{2}}}-\frac{{\left({\displaystyle \sum _{\text{i=1}}^{\text{n}}{\text{x}}_{\text{i}}}\right)}^{\text{2}}}{\text{n}}\right]\\ =\frac{1}{15-1}\left[15,524-\frac{{(472)}^2}{15}\right]=47.98\\ {\text{b}}_{\text{1}}\text{=}\frac{{\text{s}}_{\text{xy}}}{{\text{s}}_{\text{x}}^{\text{2}}} =\frac{46.42}{47.98}=0.9675\\ \overline{\text{x}}=\frac{{\displaystyle \sum {\text{x}}_{\text{i}}}}{\text{n}}\text{=}\frac{472}{\text{15}}=31.47 \\ \overline{y}=\frac{{\displaystyle \sum y_i}}{n}=\frac{86}{15}=5.73\\ {\text{b}}_{\text{0}}=\overline{\text{y}}-{\text{b}}_{\text{1}}\overline{\text{x}}\\ =5.73-\left(0.9675\right)\left(3147\right)\\ =-24.72\\ \widehat{\text{y}}=-\text{24}\text{.72}+\text{0}\text{.9675x}\end{array}$

The least square line shows that weight increases on average by 0.9675 pounds for each additional hours of television.