For the accompanying data​ set, (a) draw a scatter diagram of the​ data, (b) by​ hand, compute the correlation​ coefficient, and​ (c) determine whether there is a linear relation between x and y.
Click here to view the data set. LOADING...  
Click here to view the critical values table. LOADING...
Question content area bottom
Part 1
​(a) Draw a scatter diagram of the data. Choose the correct graph below.
A.
0
10
0
20
x
y
 


A scatter diagram has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. The following 5 points are plotted, listed here from left to right: (2, 4); (4, 8); (7, 10); (7, 7); (9, 20).
B.
0
10
0
20
x
y
 


A scatter diagram has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 2. The following 5 points are plotted, listed here from left to right: (2, 4); (4, 8); (7, 10); (7, 12); (9, 20).
This is the correct answer.C.
0
10
0
20
x
y
 


A scatter diagram has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 2. The following 5 points are plotted, listed here from left to right: (2, 20); (4, 12); (7, 10); (7, 8); (9, 4).
Your answer is not correct.D.
0
10
0
20
x
y
 


A scatter diagram has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 1. The following 5 points are plotted, listed here from left to right: (2, 20); (4, 7); (7, 10); (7, 8); (9, 4).
Part 2
​(b) By​ hand, compute the correlation coefficient.
The correlation coefficient is requals=enter your response here. ​(Round to three decimal places as​ needed.)

To solve this problem, we will tackle it step-by-step:

### Part (a): Draw a scatter diagram of the data

The data points are given as:

- (2, 4)
- (4, 8)
- (7, 10)
- (7, 7)
- (9, 20)

We need to compare the given graphs with this data set. Let's break down the options:

- **Option A:** Matches with points (2, 4), (4, 8), (7, 10), (7, 7), but doesn't include (9, 20) correctly.
- **Option B:** Matches exactly with all points.
- **Option C and D:** Both show points that don’t match the provided data set (C and D include (2, 20) and other incorrect points).

Thus, **Option B** is the correct scatter diagram.

---

### Part (b): Compute the Correlation Coefficient

We will use the formula for Pearson's correlation coefficient, \( r \), which is:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}}
\]

Where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the values of the variables.

#### Step 1: Calculate the necessary sums:

| x  | y  | x\*y | x²  | y²  |
|----|----|------|-----|-----|
| 2  | 4  | 8    | 4   | 16  |
| 4  | 8  | 32   | 16  | 64  |
| 7  | 10 | 70   | 49  | 100 |
| 7  | 7  | 49   | 49  | 49  |
| 9  | 20 | 180  | 81  | 400 |

Now, sum each column:

\[
\sum x = 2 + 4 + 7 + 7 + 9 = 29
\]
\[
\sum y = 4 + 8 + 10 + 7 + 20 = 49
\]
\[
\sum xy = 8 + 32 + 70 + 49 + 180 = 339
\]
\[
\sum x^2 = 4 + 16 + 49 + 49 + 81 = 199
\]
\[
\sum y^2 = 16 + 64 + 100 + 49 + 400 = 629
\]

#### Step 2: Apply the values into the formula:

\[
r = \frac{5(339) - (29)(49)}{\sqrt{[5(199) - (29)^2][5(629) - (49)^2]}}
\]

Simplify the expression:

\[
r = \frac{1695 - 1421}{\sqrt{[995 - 841][3145 - 2401]}}
\]
\[
r = \frac{274}{\sqrt{[154][744]}}
\]
\[
r = \frac{274}{\sqrt{114576}}
\]
\[
r = \frac{274}{338.57} \approx 0.809
\]

Thus, the correlation coefficient \( r \) is approximately **0.809**.

---

### Part (c): Determine if there is a linear relationship

Since the correlation coefficient \( r = 0.809 \) is close to 1, this indicates a strong positive linear relationship between \( x \) and \( y \).

---

Would you like more details on any part of the solution, or have any questions?

Here are 5 follow-up questions you can explore:
1. What is the interpretation of the correlation coefficient \( r \)?
2. How can the scatter diagram visually represent correlation strength?
3. What are the limitations of Pearson's correlation?
4. How does changing one data point affect the correlation coefficient?
5. What are alternative methods for determining non-linear relationships?

**Tip:** Always check if a scatter plot suggests any outliers before interpreting correlation, as they can strongly influence the result.

For the accompanying data set, (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y. Click here to view the data set. LOADING... Click here to view the critical values table. LOADING...

Learn how to analyze a dataset by drawing a scatter plot and computing the correlation coefficient. This detailed guide walks through Pearson's correlation coefficient calculation with real data points, helping determine if a linear relationship exists. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation