Chapter 4: Q. 4.134 (page 189)

In exercises, 4.134 - 4.143, we repeat data from exercises in section 4.2. For each section, determine the linear correlation coefficient by using,
(a) Definition 4.8 on page 183.
(b) Definition 4.5 on page 185
Compare your results in part (a) and (b).

Short Answer

Expert verified

(a) The correlation coefficient is0.5.

(b)The correlation coefficient is0.5.

Step by step solution

Part (a) Step 1. Given Information.

A table of values.

Part (a) Step 2. Construct the table.

Construct the table using the given values:

Part (a) Step 3. Find the mean and standard deviation.

The mean value of x is,

$\bar{x} = \frac{\sum_{} x}{n} = \frac{2 + 4 + 3}{3} = 3 \bar{y} = \frac{\sum_{} y}{n} = \frac{3 + 5 + 7}{3} = 5$

The standard deviation is given by,

$s_{x} = \sqrt{\frac{1}{n - 1} \sum_{} {(x_{i} - \bar{x})}^{2}} = \sqrt{\frac{1}{3 - 1} \sum_{} {(x_{i} - \bar{x})}^{2}} = \sqrt{1} = 1 s_{x} = \sqrt{\frac{1}{n - 1} \sum_{} {(y_{i} - \bar{y})}^{2}} = \sqrt{\frac{1}{3 - 1} \sum_{} {(y_{i} - \bar{y})}^{2}} = \sqrt{4} = 2$

Part (a) Step 4. Find the correlation coefficient.

$r = \frac{\frac{1}{n - 1} \sum_{} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{s_{x} s_{y}} = \frac{1}{2} = 0.5$

Part (b) Step 1. Construct the table.

The table is given by:

Part (b) Step 2. Find the correlation coefficient.

From the table,

$r = \frac{\sum_{} x_{i} y_{i} - \frac{\sum_{} x_{i} \underset{}{\sum y_{i}}}{n}}{\sqrt{[{x_{i}}^{2} - \frac{\underset{}{(\sum} x_{i})^{2}}{n}] [{y_{i}}^{2} - \frac{\underset{}{(\sum} y_{i})^{2}}{n}]}} = 0.5$

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In exercises, 4.134 - 4.143, we repeat data from exercises in section 4.2. For each section, determine the linear correlation coefficient by using,
(a) Definition 4.8 on page 183.
(b) Definition 4.5 on page 185
Compare your results in part (a) and (b).

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Construct the table.

Part (a) Step 3. Find the mean and standard deviation.

Part (a) Step 4. Find the correlation coefficient.

Part (b) Step 1. Construct the table.

Part (b) Step 2. Find the correlation coefficient.

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In exercises, 4.134 - 4.143, we repeat data from exercises in section 4.2. For each section, determine the linear correlation coefficient by using,(a) Definition 4.8 on page 183.(b) Definition 4.5 on page 185Compare your results in part (a) and (b).

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Construct the table.

Part (a) Step 3. Find the mean and standard deviation.

Part (a) Step 4. Find the correlation coefficient.

Part (b) Step 1. Construct the table.

Part (b) Step 2. Find the correlation coefficient.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Pure Maths

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In exercises, 4.134 - 4.143, we repeat data from exercises in section 4.2. For each section, determine the linear correlation coefficient by using,
(a) Definition 4.8 on page 183.
(b) Definition 4.5 on page 185
Compare your results in part (a) and (b).