Q. 4.54 4.54 The data points in Exercise... [FREE SOLUTION]

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Chapter 4: Q. 4.54 (page 174)

4.54 The data points in Exercise $4.42$
a. find the regression equotion for the data points. Use the defining formulas in Definition $4.4$ to obtain $S_{x x}$ and $S_{x y}$ .
b. graph the regression equation and the data points.

Short Answer

Expert verified

(a) The regression equation is $\begin{aligned} \hat{y} = 9 - 2 x \end{aligned}$ .

(b) the graph for the regression equation is

Step by step solution

Part (a) Step 1: Given information

The data points in Exercise $4.42$ is:

$x$	$1$	$3$	$4$	$4$
$y$	$8$	$0$	$3$	$1$

Part (a) Step 2: Explanation

The terms $b_{0}$ and $b_{1}$ are used to generate the regression equation.
For a set of $n$ data points, the regression equation is:
$\hat{y} = b_{0} + b_{1} x$
Note that,
$b_{1} = \frac{S_{x y}}{S_{x x}} b_{0} = \frac{1}{n} (鈭� y_{i} 鈭� b_{1} 鈭� x_{i}) b_{0} = \overset{炉}{y} 鈭� b_{1} \overset{炉}{x}$

The table is available in the following formats:

$x$	$y$	$(x_{i} - \overset{炉}{x})$	${(x_{i} - \overset{炉}{x})}^{2}$	$(y_{i} - \overset{炉}{y})$	localid="1650867898682" $(x_{i} - \overset{炉}{x}) (y_{i} - \overset{炉}{y})$
$1$	$8$	$- 2$	$4$	$5$	$- 10$
$3$	$0$	$0$	$0$	$- 3$	$0$
$4$	$3$	$1$	$1$	$0$	$0$
$4$	$1$	$1$	$1$	$- 2$	$- 2$
$\overset{炉}{x} = 3$	$\overset{炉}{y} = 3$		localid="1650867836278" $鈭� {(x_{i} - \overset{炉}{x})}^{2} = 6$		localid="1650867920723" $鈭� (x_{i} - \overset{炉}{x}) (y_{i} - \overset{炉}{y}) = - 12$

Hence, $S_{x x} = 6$ and $S_{x y} = - 12$ .

Part (a) Step 3: Calculation

Since,
$b_{1} = \frac{S_{x y}}{S_{x x}} b_{1} = \frac{- 12}{6} b_{1} = - 2$

Then,

$b_{0} = \overset{炉}{y} 鈭� b_{1} \overset{炉}{x} b_{0} = 3 鈭� (鈭� 2) 3 b_{0} = 3 + 6 b_{0} = 9$

As a result, the regression equation will be as follows:

$\begin{aligned} \hat{y} = b_{0} + b_{1} x \end{aligned} \begin{aligned} \hat{y} = 9 + (鈭� 2) x \end{aligned} \begin{aligned} \hat{y} = 9 - 2 x \end{aligned}$

Part (b) Step 1: Given information

Given in the question that

Part (b) Step 2: Graphical representation

The graph for the regression equation $\hat{y} = 9 鈭� 2 x$ is

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91影视

4.54 The data points in Exercise $4.42$
a. find the regression equotion for the data points. Use the defining formulas in Definition $4.4$ to obtain $S_{x x}$ and $S_{x y}$ .
b. graph the regression equation and the data points.

Short Answer

Step by step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

Part (a) Step 3: Calculation

Part (b) Step 1: Given information

Part (b) Step 2: Graphical representation

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91影视

4.54 The data points in Exercise4.42a. find the regression equotion for the data points. Use the defining formulas in Definition 4.4to obtain Sxxand Sxy.b. graph the regression equation and the data points.

Short Answer

Step by step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

Part (a) Step 3: Calculation

Part (b) Step 1: Given information

Part (b) Step 2: Graphical representation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Logic and Functions

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

4.54 The data points in Exercise $4.42$
a. find the regression equotion for the data points. Use the defining formulas in Definition $4.4$ to obtain $S_{x x}$ and $S_{x y}$ .
b. graph the regression equation and the data points.