Q. 4.56 - Elementary Statistics

Question

 Data points: 

X	1	1	5	5
Y	1	3	2	4

• a. Find the regression equation for the data points. Use the defining formulas in Definition 4.4 to obtain $S_{xx}$ and $S_{xy}$ .
• b. graph the regression equation and the data points.

Step-by-Step Solution

Answer

Ans: 

part (a): The regression equation for the given data $\hat{y}=b_0+b_{1}x =1.75+0.25x$

part (b): Graph representation

Step 1. Given information:

Data points: 

X	1	1	5	5
Y	1	3	2	4

Step 2. Solving part (a):

Obtain $S_{xx}$ and $S_{xy}$ .

The formula for $S_{xx}$ and $S_{xy}$ is given below.

$$\begin{gathered} S_{xx}=\sum {\left(x_i-\overline{x}\right)}^2 \\ {S}_{xy}=\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right) \end{gathered}$$

Therefore, $\sum {\left(x_i-\overline{x}\right)}^2$ and $\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)$ are tabulated below:

From the table, the value of $S_{xx}$ is 16, and the value of $S_{xy}$ is 4 .v

Step 3. Continue:

Find the slope.

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ =\frac{4}{16} \\ =0.25 \end{gathered}$$

Find the y-intercept.

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ =2.5-(0.25)\left(3\right) \\ =2.5-0.75 \\ =1.75 \end{gathered}$$

Therefore, the regression equation for the given data is,

$$\begin{gathered} \hat{y}=b_0+b_{1}x \\ =1.75+0.25x \end{gathered}$$

Step 4. Solving part (b):

Graph the regression equation and the data points by using MINITAB.

MINITAB procedure:

Step 1: Choose Graph > Scatterplot.

Step 2: Choose With Connect Lina and then click OK.

Step 3: Under Y variables, enter a column of y.

Step 4: Under X variables, enter a column of X.

Step 5: Click OK.

Step 5. Continue: