Math Problem Statement

Estimate the coefficient of correlation using Karl Pearson formula.

Solution

To calculate the coefficient of correlation using Karl Pearson's formula, we can follow these steps:

Formula for Pearson's Correlation Coefficient (r):

[ 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:

  • XX and YY are the variables in the given data,
  • nn is the number of pairs,
  • X\sum{X}, Y\sum{Y}, X2\sum{X^2}, Y2\sum{Y^2}, and XY\sum{XY} are the sums of the corresponding terms.

Given the table:

X10121525202719
Y14172324182114

Let’s proceed step by step.

  1. Step 1: Find the sum of X, Y, X^2, Y^2, and XY.

I will calculate these values to help compute the correlation coefficient. numpy as np

Given data

X = np.array([10, 12, 15, 25, 20, 27, 19]) Y = np.array([14, 17, 23, 24, 18, 21, 14])

Calculate the required sums

n = len(X) sum_X = np.sum(X) sum_Y = np.sum(Y) sum_X_squared = np.sum(X2) sum_Y_squared = np.sum(Y2) sum_XY = np.sum(X * Y)

(sum_X, sum_Y, sum_X_squared, sum_Y_squared, sum_XY)The calculated sums are:

  • X=128\sum{X} = 128
  • Y=131\sum{Y} = 131
  • X2=2584\sum{X^2} = 2584
  • Y2=2551\sum{Y^2} = 2551
  • XY=2482\sum{XY} = 2482

Step 2: Apply the Pearson's Correlation Coefficient Formula

Now, we can substitute these values into the formula:

[ 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=7n = 7.

Let's compute the correlation coefficient rr. Compute Pearson's correlation coefficient r_numerator = n * sum_XY - sum_X * sum_Y r_denominator = np.sqrt((n * sum_X_squared - sum_X2) * (n * sum_Y_squared - sum_Y2))

r = r_numerator / r_denominator rThe Pearson correlation coefficient (rr) is approximately 0.556.

This indicates a moderate positive linear relationship between the variables XX and YY.

Would you like further clarification on this calculation or have any additional questions?

Here are some related questions you may find useful:

  1. How can you interpret different values of the Pearson correlation coefficient?
  2. What are some assumptions behind Pearson’s correlation?
  3. How can Pearson’s correlation be used in data analysis?
  4. What are the limitations of Pearson’s correlation coefficient?
  5. How do you calculate the coefficient of determination (r²)?

Tip: Always check for linearity and normality of the data before using Pearson’s correlation, as the formula assumes these conditions.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Karl Pearson's Coefficient

Formulas

Pearson's Correlation Coefficient: r = [nΣXY - ΣXΣY] / √{[nΣX^2 - (ΣX)^2][nΣY^2 - (ΣY)^2]}

Theorems

Karl Pearson's Correlation Coefficient Formula

Suitable Grade Level

Undergraduate level or high school statistics