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: Find the mean of x and y

Let $n = 18$.
$\bar{x}=\frac{1.6 + 3.7+8.6+5.2+9.2+4.3+10.3+2.1+7.2+8.2+4.8+6.5+9.7+2.5+6.8+5.9+2.8+3.8}{18}=\frac{103.7}{18}\approx5.76$
$\bar{y}=\frac{3.7+3.7+7.0+7.0+8.2+4.4+7.9+3.9+7.0+8.2+5.1+5.9+7.8+2.9+5.7+7.8+3.9+5.4}{18}=\frac{107.6}{18}\approx5.98$

Step2: Calculate the slope (b1)

$b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$
First, calculate $(x_i-\bar{x})(y_i - \bar{y})$ and $(x_i-\bar{x})^2$ for each $i$ and sum them up.
After calculations, $\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\approx77.79$ and $\sum_{i=1}^{n}(x_i-\bar{x})^2\approx84.97$
$b_1=\frac{77.79}{84.97}\approx0.92$

Step3: Calculate the intercept (b0)

$b_0=\bar{y}-b_1\bar{x}$
$b_0 = 5.98-0.92\times5.76=5.98 - 5.30=0.68$

Step4: Sketch the line

The least - squares regression line is $y = b_0 + b_1x=0.68+0.92x$. To sketch it, find two points on the line. For example, when $x = 0$, $y=0.68$ and when $x = 10$, $y=0.68 + 0.92\times10=9.88$. Plot these two points and draw a straight line through them on the scatter - plot.

Answer:

Sketch the line $y = 0.68+0.92x$ on the given scatter - plot using two points (0, 0.68) and (10, 9.88)