Correlation Coefficient, Regression, and Chebyshev's Theorem

Calculating Karl Pearson Correlation Coefficient

Question 11: Calculate the coefficient of correlation from the following data:

To calculate Karl Pearson's coefficient of correlation (r), we use the formula:

r = ∑dx·dy / √(∑dx² × ∑dy²)

Where:

• dx = X - X̄
• dy = Y - Ȳ

Step 1: Find the Means

First, calculate the mean of X (X̄) and Y (Ȳ):

X̄ = (12 + 9 + 8 + 10 + 11 + 13 + 7) / 7 = 70 / 7 = 10

Ȳ = (14 + 8 + 6 + 9 + 11 + 12 + 3) / 7 = 63 / 7 = 9

Step 2: Prepare the Calculation Table

Step 3: Substitute into the Formula

Substitute the sums from the table into the correlation formula:

r = 46 / √(28 × 84)
r = 46 / √(2352)
r = 46 / 48.50 ≈ 0.949

Answer

The coefficient of correlation is r ≈ 0.95.

Proving Regression Line Coefficients Range

Question 12: Given that x = 4y + 5 and y = kx + 4 are the regression lines of x on y and y on x respectively, show that 0 ≤ k ≤ 0.25.

Solution

The regression coefficients are:

• Regression coefficient of x on y: b_xy = 4
• Regression coefficient of y on x: b_yx = k

The product of the two regression coefficients is equal to the square of the correlation coefficient:

b_xy × b_yx = r²

Hence,

4k = r²

Since the correlation coefficient satisfies -1 ≤ r ≤ 1, we have:

0 ≤ r² ≤ 1

Therefore,

0 ≤ 4k ≤ 1

Dividing throughout by 4, we get:

0 ≤ k ≤ 1/4 or 0 ≤ k ≤ 0.25.

State and Prove Chebyshev's Theorem

Question 6: State and prove Chebyshev's Theorem.

Proof

Let X be a random variable with mean μ and variance σ².

By Markov's Inequality, for any non-negative random variable:

P((X - μ)² ≥ k²σ²) ≤ E[(X - μ)²] / (k²σ²)

Since E[(X - μ)²] = σ², we get:

P((X - μ)² ≥ k²σ²) ≤ σ² / (k²σ²) = 1 / k²

But we know that:

(X - μ)² ≥ k²σ² ⇔ |X - μ| ≥ kσ

Hence,

P(|X - μ| ≥ kσ) ≤ 1 / k²

Taking the complement of the probability:

P(|X - μ| < kσ) = 1 - P(|X - μ| ≥ kσ) ≥ 1 - 1 / k²

Thus,

P(|X - μ| < kσ) ≥ 1 - 1 / k²

Hence, Chebyshev's Theorem is proved.

Important Results

• Within 2 standard deviations (k = 2): 1 - 1/2² = 3/4 = 75%
• Within 3 standard deviations (k = 3): 1 - 1/3² = 8/9 &approx; 88.89%