Math Problem Statement
Based on the data shown below, calculate the correlation coefficient (rounded to three decimal places) | x | y | |----|-------| | 4 | 36.2 | | 5 | 35.78 | | 6 | 33.56 | | 7 | 31.34 | | 8 | 27.32 | | 9 | 24.5 | | 10 | 22.18 | | 11 | 20.76 | | 12 | 16.34 | | 13 | 14.82 | | 14 | 13.7 |
Solution
To calculate the correlation coefficient (), we use the formula:
[ r = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{\sqrt{\left[n\sum{x^2} - (\sum{x})^2\right] \left[n\sum{y^2} - (\sum{y})^2\right]}} ]
Where:
- and are the variables.
- is the number of data points.
- , , , , and are calculated based on the data.
Step 1: Compute Necessary Sums
Let’s calculate:
4 | 36.2 | 16 | 1310.44 | 144.8 |
5 | 35.78 | 25 | 1280.2084 | 178.9 |
6 | 33.56 | 36 | 1126.8736 | 201.36 |
7 | 31.34 | 49 | 982.5956 | 219.38 |
8 | 27.32 | 64 | 746.1424 | 218.56 |
9 | 24.5 | 81 | 600.25 | 220.5 |
10 | 22.18 | 100 | 492.7524 | 221.8 |
11 | 20.76 | 121 | 431.5776 | 228.36 |
12 | 16.34 | 144 | 267.6356 | 196.08 |
13 | 14.82 | 169 | 219.5024 | 192.66 |
14 | 13.7 | 196 | 187.69 | 191.8 |
Now sum each column:
Step 2: Plug Values into the Correlation Formula
[ r = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{\sqrt{\left[n\sum{x^2} - (\sum{x})^2\right] \left[n\sum{y^2} - (\sum{y})^2\right]}} ]
Substitute:
- ,
- ,
- ,
- ,
- ,
- .
Numerator:
Denominator:
-
First term: :
-
Second term: :
-
Denominator product:
Correlation Coefficient:
Final Answer
The correlation coefficient is:
This indicates a weak negative correlation between and . Let me know if you'd like further clarification!
Related Questions
- What does the sign of the correlation coefficient indicate about the relationship between two variables?
- How would outliers affect the value of the correlation coefficient?
- What are some real-world examples of weak negative correlations?
- Can the correlation coefficient determine causation? Why or why not?
- How can we test the significance of a correlation coefficient?
Tip: Always visualize data (e.g., scatter plots) when interpreting correlation coefficients to spot patterns or outliers!
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Math Problem Analysis
Mathematical Concepts
Statistics
Correlation Coefficient
Descriptive Statistics
Formulas
Correlation coefficient formula: r = (n∑xy - ∑x∑y) / sqrt[(n∑x² - (∑x)²)(n∑y² - (∑y)²)]
Theorems
Pearson's correlation coefficient theorem
Suitable Grade Level
Grades 9-12