The data below shows the high temperatures and the times​ (in minutes) runners who won a marathon.	
Temperature​ (x)
	
59
65
49
60
75
72
50
57                                                                                    
Time​ (y)
	
145.067
145.706
147.073
144.657
144.921
146.627
147.925
144.376.                                                                                 Find the value of the linear correlation coefficient r.

  

To find the linear correlation coefficient \( r \) between temperature \( x \) and time \( y \), we can use the formula for Pearson's correlation coefficient:

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

Where:
- \( n \) is the number of data points
- \( \sum x \) is the sum of the temperatures
- \( \sum y \) is the sum of the times
- \( \sum xy \) is the sum of the product of corresponding \( x \) and \( y \) values
- \( \sum x^2 \) is the sum of the squares of the temperatures
- \( \sum y^2 \) is the sum of the squares of the times

I'll compute this for you.The linear correlation coefficient \( r \) is approximately \(-0.403\), indicating a moderate negative correlation between temperature and marathon winning time. This suggests that as the temperature increases, the winning time tends to slightly decrease.

Would you like further details on how this was calculated or how to interpret the result? Here are 5 related questions to consider:

1. How is the linear correlation coefficient used in data analysis?
2. What does a positive or negative value of \( r \) indicate?
3. How does temperature affect marathon performance based on this data?
4. What is the significance of the value of \( r \) being close to zero, positive, or negative?
5. How can we improve the accuracy of correlation analysis?

**Tip:** The linear correlation coefficient only measures linear relationships. For non-linear relationships, other methods may be more appropriate.

The data below shows the high temperatures and the times​ (in minutes) runners who won a marathon. Temperature​ (x): 59, 65, 49, 60, 75, 72, 50, 57. Time​ (y): 145.067, 145.706, 147.073, 144.657, 144.921, 146.627, 147.925, 144.376. Find the value of the linear correlation coefficient r.

Calculate the linear correlation coefficient (r) between high temperatures and marathon winning times using Pearson's correlation formula. This solution demonstrates how temperature affects marathon performance, providing a moderate negative correlation of approximately -0.403. Suitable for high school students in Grades 10-12 studying statistics and data analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation