Q.4.47 - Elementary Statistics

Question

For a data set consisting of 2 data points:

For each of the following sets of data points, determine the regression equation both without and with the use of formula 4.1 on page 165.

Step-by-Step Solution

Answer

(a) with formula:  $\hat{y}=-1+x$

   without formula: $\hat{y}=-1.00+1.00x$

(b) with formula : $\hat{y}=4.5-1.5x$

   without formula : $\hat{y}=4.50-1.50x$

Part (a) Step 1 :Given Information

The given data set is:

we have to determine the regression equation both without and with  using formula.

Part (a) Step 2: Explanation

The regression equation is obtained by using the terms $b_{0 }and b_1$ The regression equation for a set of $n$ data points is

$\hat{y}=b_0+b_{1}x$

where,

 $b_1=\frac{s_{xy}}{s_{xx}}$

$b_0=\overline{y}-b_1\overline{x}$

The table can be obtained as:

$\overline{x}=3$

$\overline{y}=2$

$$\begin{gathered} s_{xx}=2 \\ {s}_{xy}=2 \end{gathered}$$

therefore,

$$\begin{gathered} b_1=1 \\ {b}_0=-1 \end{gathered}$$

therefore the regression equation will be:

$\hat{y}=-1+x$

The regression equation without using the formula is:

Thus, the regression equation will be$\hat{y}=-1.00+1.00x$

Part (b) Step 1 :Given Information

The given data is:

we have to find the regression equation with or without formula.

Part (b) Step 2: Explanation

The regression equation is obtained by using the terms  $b_{0 }and b_1$The regression equation for a set of  $n$data points is

$\hat{y}=b_0+b_{1}x$

where,

$b_1=\frac{s_{xy}}{s_{xx}}$

$b_0=\overline{y}-b_1\overline{x}$

The table can be obtained as:

$\overline{x}=3$

$\overline{Y}=0$

$s_{xx}=8$

$s_{xy}=-12$

Therefore,

$b_1=-1.5$

$b_0=4.5$

Therefore the regression equation will be:

$\hat{y}=4.5-1.5x$

The regression equation without using the formula is:

Thus, the regression equation will be$\hat{y}=4.50-1.50x$