Q. 4.55 4.55 The data points in Exercise... [FREE SOLUTION]

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Chapter 4: Q. 4.55 (page 174)

4.55 The data points in Exercise 4.43
a. find the regression equation for the data points. Use the defining formations in Definition $4.4$ to obtain $S_{x x}$ and $S_{x y}$
b. Graph the regression connection and the data points.

Short Answer

Expert verified

(a) The regression equation is $\hat{y} = 1 + 2 x$ ,

(b) The graph will be

Step by step solution

Part (a) Step 1: Given information

The data points in Exercise 4.43 is:

$x$	$1$	$2$	$3$
$y$	$4$	$3$	$8$

Part (a) Step 2: Explanation

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

The table is available in the following formats:

$x$	$y$	$鈭� (x_{i} 鈭� \overset{炉}{x})$	$鈭� {(x_{i} 鈭� \overset{炉}{x})}^{2}$	$(y_{i} - y)$	$(x_{i} 鈭� \overset{炉}{x}) (y_{i} - y)$
$1$	$4$	$- 1$	$1$	$- 1$	$1$
$2$	$3$	$0$	$0$	$- 2$	$0$
$3$	$8$	$1$	$1$	$3$	$3$
$x = 2$	$y = 5$		$鈭� {(x_{i} 鈭� \overset{炉}{x})}^{2} = 2$		$鈭� (x_{i} 鈭� \overset{炉}{x}) (y_{i} - y) = 4$

Hence, $S_{x x} = 2$ , and $S_{x y} = 4$ .

Part (a) Step 3: Claculation

Since,

$b_{1} = \frac{S_{x y}}{S_{x x}} b_{1} = \frac{4}{2} b_{1} = 2$

Then,

$b_{0} = \overset{炉}{y} 鈭� b_{1} \overset{炉}{x} b_{0} = 5 - (2) 2 b_{0} = 5 - 4 b_{0} = 1$

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

$\hat{y} = b_{0} + b_{1} x \hat{y} = 1 + (2) x \hat{y} = 1 + 2 x$

Part (b) Step 1: Given information

Given in the question that

Part (b) Step 2: Graphical representation

The graph for the regression equation $\hat{y} = 1 + 2 x$ is

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91影视

4.55 The data points in Exercise 4.43
a. find the regression equation for the data points. Use the defining formations in Definition $4.4$ to obtain $S_{x x}$ and $S_{x y}$
b. Graph the regression connection 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: Claculation

Part (b) Step 1: Given information

Part (b) Step 2: Graphical representation

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91影视

4.55 The data points in Exercise 4.43a. find the regression equation for the data points. Use the defining formations in Definition 4.4to obtain Sxxand Sxyb. Graph the regression connection 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: Claculation

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

Decision Maths

Geometry

Discrete Mathematics

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

4.55 The data points in Exercise 4.43
a. find the regression equation for the data points. Use the defining formations in Definition $4.4$ to obtain $S_{x x}$ and $S_{x y}$
b. Graph the regression connection and the data points.