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 (rr), 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:

  • xx and yy are the variables.
  • nn is the number of data points.
  • x\sum{x}, y\sum{y}, xy\sum{xy}, x2\sum{x^2}, and y2\sum{y^2} are calculated based on the data.

Step 1: Compute Necessary Sums

Let’s calculate:

  1. x\sum{x}
  2. y\sum{y}
  3. xy\sum{xy}
  4. x2\sum{x^2}
  5. y2\sum{y^2}
xxyyx2x^2y2y^2xyxy
436.2161310.44144.8
535.78251280.2084178.9
633.56361126.8736201.36
731.3449982.5956219.38
827.3264746.1424218.56
924.581600.25220.5
1022.18100492.7524221.8
1120.76121431.5776228.36
1216.34144267.6356196.08
1314.82169219.5024192.66
1413.7196187.69191.8

Now sum each column:

x=99,y=276.2,x2=1001,y2=7645.268,xy=2214.16\sum{x} = 99, \quad \sum{y} = 276.2, \quad \sum{x^2} = 1001, \quad \sum{y^2} = 7645.268, \quad \sum{xy} = 2214.16


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:

  • n=11n = 11,
  • xy=2214.16\sum{xy} = 2214.16,
  • x=99\sum{x} = 99,
  • y=276.2\sum{y} = 276.2,
  • x2=1001\sum{x^2} = 1001,
  • y2=7645.268\sum{y^2} = 7645.268.

Numerator:

n(xy)xy=11(2214.16)(99)(276.2)=24355.7627343.8=988.04n\sum{(xy)} - \sum{x}\sum{y} = 11(2214.16) - (99)(276.2) = 24355.76 - 27343.8 = -988.04

Denominator:

  1. First term: nx2(x)2n\sum{x^2} - (\sum{x})^2: 11(1001)(99)2=110119801=121011(1001) - (99)^2 = 11011 - 9801 = 1210

  2. Second term: ny2(y)2n\sum{y^2} - (\sum{y})^2: 11(7645.268)(276.2)2=84097.94876280.44=7817.50811(7645.268) - (276.2)^2 = 84097.948 - 76280.44 = 7817.508

  3. Denominator product: 12107817.508=9463160.68=3076.05\sqrt{1210 \cdot 7817.508} = \sqrt{9463160.68} = 3076.05

Correlation Coefficient:

r=988.043076.05=0.321r = \frac{-988.04}{3076.05} = -0.321


Final Answer

The correlation coefficient is: 0.321\boxed{-0.321}

This indicates a weak negative correlation between xx and yy. Let me know if you'd like further clarification!


Related Questions

  1. What does the sign of the correlation coefficient indicate about the relationship between two variables?
  2. How would outliers affect the value of the correlation coefficient?
  3. What are some real-world examples of weak negative correlations?
  4. Can the correlation coefficient determine causation? Why or why not?
  5. 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