Math Problem Statement
Compute r, the correlation coefficient, using the following data. x 7 1 3 8 2 5
y 5 9 9 3 9 7
Solution
To calculate the correlation coefficient between the two variables and , we will 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:
- is the number of data points.
- is the sum of the -values.
- is the sum of the -values.
- is the sum of the product of corresponding and values.
- is the sum of the squares of the -values.
- is the sum of the squares of the -values.
Step-by-step Calculation:
Given the data:
Let's calculate the necessary sums:
-
Sum of :
-
Sum of :
-
Sum of :
-
Sum of :
-
Sum of :
Now, we can plug these values into the formula:
[ r = \frac{6(148) - (26)(42)}{\sqrt{[6(152) - (26)^2][6(326) - (42)^2]}} ]
[ r = \frac{888 - 1092}{\sqrt{[912 - 676][1956 - 1764]}} ]
[ r = \frac{-204}{\sqrt{[236][192]}} ]
Final Answer:
The correlation coefficient is approximately -0.96, which indicates a very strong negative linear relationship between and .
Would you like a more detailed breakdown or have any questions about the process?
Here are some related questions you might find interesting:
- How does the value of the correlation coefficient relate to the strength and direction of the relationship between two variables?
- What would happen to the correlation coefficient if the data points were reversed (i.e., switching and )?
- How do you interpret a correlation coefficient of -1 or 1?
- What are other ways to assess the relationship between two variables besides the correlation coefficient?
- Can the correlation coefficient be used for non-linear relationships?
Tip: If you get a very high or very low correlation coefficient (close to 1 or -1), double-check the data for outliers or patterns that might be influencing the result too strongly!
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Math Problem Analysis
Mathematical Concepts
Statistics
Correlation Coefficient
Pearson's Correlation
Formulas
r = (n(Σxy) - (Σx)(Σy)) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]
Theorems
Pearson's Correlation Coefficient Theorem
Suitable Grade Level
Grades 9-12
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