Linear Regression and Correlation.

x	y
3	0.73
4	8.54
5	9.65
6	6.46
7	5.87
8	3.38
9	4.09
10	5.8
11	4.71
12	7.52
13	6.13
14	7.34
15	12.25
16	10.76
17	13.17
18	10.68
19	8.29
20	13.5

Using your calculator, compute the equation of the linear regression line in the form 
, where 
 is the intercept and 
 is the slope.

Use at least 3 decimal places. (Round if necessary)
 = 
 + 
x

Compute the correlation coefficient for this data set. Use at least 3 decimal places. (Round if necessary)
r= 


Is the correlation significant? Use the table of critical values to test.
No
Yes, significant correlation

The equation of the linear regression line is:

\[
y = 2.651 + 0.440x
\]

The correlation coefficient \( r \) is approximately 0.673.

### Testing the significance of the correlation:
To determine if the correlation is significant, you can compare the absolute value of the correlation coefficient against a critical value from a correlation table for \( n = 18 \) data points at a given significance level (e.g., 0.05). For \( n = 18 \), the critical value at \( \alpha = 0.05 \) is approximately 0.468.

Since \( |r| = 0.673 \) is greater than 0.468, the correlation is significant.

Thus, the answer is: **Yes, there is a significant correlation.**

Would you like further details or clarifications?

Here are 5 related questions you could explore:
1. How does the p-value relate to the significance of the correlation?
2. What happens if we add more data points to this dataset?
3. How do outliers affect the correlation coefficient?
4. What other forms of regression could be applied to this dataset?
5. How does the standard error influence the accuracy of our regression equation?

**Tip**: Always check if assumptions for linear regression (such as linearity and homoscedasticity) hold for the data.

Using the following data set, compute the equation of the linear regression line and the correlation coefficient:

x  |  y
---|---
3  | 0.73
4  | 8.54
5  | 9.65
6  | 6.46
7  | 5.87
8  | 3.38
9  | 4.09
10 | 5.80
11 | 4.71
12 | 7.52
13 | 6.13
14 | 7.34
15 | 12.25
16 | 10.76
17 | 13.17
18 | 10.68
19 | 8.29
20 | 13.50

Determine if the correlation is significant.

Learn how to compute the linear regression equation and correlation coefficient for a given data set, including values such as x = [3, 4, 5,..., 20] and y = [0.73, 8.54,..., 13.5]. This problem explains the linear regression formula, the significance test for the correlation coefficient, and its interpretation. Suitable for high school students in Grades 10-12.

Math Problem Statement

Use at least 3 decimal places. (Round if necessary)

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