Critical Values for Correlation Coefficient

n

3

0.997

4

0.950

5

0.878

6

0.811

7

0.754

8

0.707

9

0.666

10

0.632

11

0.602

12

0.576

13

0.553

14

0.532

15

0.514

16

0.497

17

0.482

18

0.468

19

0.456

20

0.444

21

0.433

22

0.423

23

0.413

24

0.404

25

0.396

26

0.388

27

0.381

28

0.374

29

0.367

30

0.361

n                                          data set x	y

2	4

4	8

6	12

6	13

7	19​(b) By​ hand, compute the correlation coefficient.

The correlation coefficient is 

requals=enter your response here. 

​(Round to three decimal places as​ needed.)

To compute the correlation coefficient \( r \) by hand, we will use the formula for the Pearson correlation coefficient:

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

### Given Data:
\( x = [2, 4, 6, 6, 7] \)  
\( y = [4, 8, 12, 13, 19] \)  
\( n = 5 \) (since there are 5 pairs of data)

We need to calculate the following values:

- \( \sum x \)
- \( \sum y \)
- \( \sum xy \)
- \( \sum x^2 \)
- \( \sum y^2 \)

I will calculate these step by step for you.import numpy as np

# Given data
x = np.array([2, 4, 6, 6, 7])
y = np.array([4, 8, 12, 13, 19])
n = len(x)

# Calculations
sum_x = np.sum(x)
sum_y = np.sum(y)
sum_xy = np.sum(x * y)
sum_x2 = np.sum(x ** 2)
sum_y2 = np.sum(y ** 2)

# Correlation coefficient formula
r = (n * sum_xy - sum_x * sum_y) / np.sqrt((n * sum_x2 - sum_x ** 2) * (n * sum_y2 - sum_y ** 2))
rThe correlation coefficient \( r \) is approximately \( 0.955 \) (rounded to three decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the correlation coefficient interpreted in terms of strength and direction?
2. What are the limitations of using the Pearson correlation coefficient?
3. How does the value of \( r \) relate to positive and negative association?
4. What is the difference between correlation and causation?
5. How do you compute the correlation coefficient when given larger datasets?

**Tip:** A correlation coefficient close to 1 indicates a strong positive linear relationship between variables.

Learn how to compute the correlation coefficient manually with a step-by-step guide. This problem involves calculating the Pearson correlation coefficient using given data sets. Suitable for advanced high school students or anyone interested in statistical analysis.

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