Q. 4.54 - Elementary Statistics

Question

 Data points:

x	1	3	4	4
y	8	0	3	1

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

Step-by-Step Solution

Answer

Ans: 

part (a): The regression equation for the given data $\hat{y}=b_0+b_{1}x =9-2x$

part (b): Graph representation

Step 1. Given information:

Data points:

x	1	3	4	4
y	8	0	3	1

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 6, and the value of $S_{xy}$ is  -12.

Step 3. Continue:

Find the slope.

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ =\frac{-12}{6} \\ =-2 \end{gathered}$$

Find the y-intercept.

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ =3-(-2)\left(3\right) \\ -3+6 \\ =9 \end{gathered}$$

Therefore, the regression equation for the given data is,

$$\begin{gathered} \hat{y}=b_0+b_{1}x \\ =9-2x \end{gathered}$$

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 Line, 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: