Question

bivariate data for the quantitative variables x and y are given in the table below. these data are plotted in the scatter plot shown next to the table. in the scatter plot, sketch an approximation of the least - squares regression line for the data.

x	y
3.7	3.7
8.6	7.0
5.2	7.0
9.2	8.2
4.3	4.4
10.3	7.9
2.1	3.9
7.2	7.0
8.2	8.2
4.8	5.1
6.5	5.9
9.7	7.8
2.5	2.9
6.8	5.7
5.9	7.8
2.8	3.9
3.8	5.4

Explanation:

Step1: Calculate means of x and y

Let \(n = 18\).
\(\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_{i}=\frac{1.6 + 3.7+8.6+5.2+9.2+4.3+10.3+2.1+7.2+8.2+4.8+6.5+9.7+2.5+6.8+5.9+2.8+3.8}{18}\)
\(\bar{x}=\frac{100.7}{18}\approx5.594\)
\(\bar{y}=\frac{1}{n}\sum_{i = 1}^{n}y_{i}=\frac{3.7 + 3.7+7.0+7.0+8.2+4.4+7.9+3.9+7.0+8.2+5.1+5.9+7.8+2.9+5.7+7.8+3.9+5.4}{18}\)
\(\bar{y}=\frac{107.6}{18}\approx5.978\)

Step2: Calculate slope \(b_1\)

\(b_1=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}\)
First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^2\) for each \(i\) and sum them up.
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\) and \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^2\) values are obtained by:
For example, for \(x_1 = 1.6,y_1 = 3.7\): \((1.6 - 5.594)(3.7- 5.978)\) and \((1.6 - 5.594)^2\)
After calculating all 18 - pairs and summing:
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\approx97.97\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^2\approx109.17\)
\(b_1=\frac{97.97}{109.17}\approx0.897\)

Step3: Calculate intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 5.978-0.897\times5.594\)
\(b_0=5.978 - 5.018\approx0.96\)
The least - squares regression line is \(y = b_0 + b_1x=0.96+0.897x\)
To sketch the line on the scatter - plot:

1. Locate the point \((\bar{x},\bar{y})\approx(5.594,5.978)\) on the scatter - plot.
2. Use the slope \(b_1 = 0.897\approx0.9\). From the point \((\bar{x},\bar{y})\), for a 1 - unit increase in \(x\), the \(y\) - value increases by approximately \(0.9\) units. Draw a straight line passing through points estimated using the slope and the point \((\bar{x},\bar{y})\).

Answer:

Sketch the line \(y = 0.96+0.897x\) on the scatter - plot using the steps above.