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 rr between the two variables xx and yy, 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:

  • nn is the number of data points.
  • x\sum x is the sum of the xx-values.
  • y\sum y is the sum of the yy-values.
  • xy\sum xy is the sum of the product of corresponding xx and yy values.
  • x2\sum x^2 is the sum of the squares of the xx-values.
  • y2\sum y^2 is the sum of the squares of the yy-values.

Step-by-step Calculation:

Given the data:

x=[7,1,3,8,2,5]x = [7, 1, 3, 8, 2, 5] y=[5,9,9,3,9,7]y = [5, 9, 9, 3, 9, 7]

Let's calculate the necessary sums:

  1. Sum of xx: x=7+1+3+8+2+5=26\sum x = 7 + 1 + 3 + 8 + 2 + 5 = 26

  2. Sum of yy: y=5+9+9+3+9+7=42\sum y = 5 + 9 + 9 + 3 + 9 + 7 = 42

  3. Sum of x2x^2: x2=72+12+32+82+22+52=49+1+9+64+4+25=152\sum x^2 = 7^2 + 1^2 + 3^2 + 8^2 + 2^2 + 5^2 = 49 + 1 + 9 + 64 + 4 + 25 = 152

  4. Sum of y2y^2: y2=52+92+92+32+92+72=25+81+81+9+81+49=326\sum y^2 = 5^2 + 9^2 + 9^2 + 3^2 + 9^2 + 7^2 = 25 + 81 + 81 + 9 + 81 + 49 = 326

  5. Sum of xyxy: xy=(7×5)+(1×9)+(3×9)+(8×3)+(2×9)+(5×7)=35+9+27+24+18+35=148\sum xy = (7 \times 5) + (1 \times 9) + (3 \times 9) + (8 \times 3) + (2 \times 9) + (5 \times 7) = 35 + 9 + 27 + 24 + 18 + 35 = 148

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]}} ]

r=20445216=204212.56r = \frac{-204}{\sqrt{45216}} = \frac{-204}{212.56}

r0.96r \approx -0.96

Final Answer:

The correlation coefficient rr is approximately -0.96, which indicates a very strong negative linear relationship between xx and yy.


Would you like a more detailed breakdown or have any questions about the process?

Here are some related questions you might find interesting:

  1. How does the value of the correlation coefficient relate to the strength and direction of the relationship between two variables?
  2. What would happen to the correlation coefficient if the data points were reversed (i.e., switching xx and yy)?
  3. How do you interpret a correlation coefficient of -1 or 1?
  4. What are other ways to assess the relationship between two variables besides the correlation coefficient?
  5. 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