Fundamentals of Mathematical Statistics (BCA) 1st Sem Previous Year Solved Question Paper 2022

To calculate the Karl Pearson's coefficient of correlation (commonly known as Pearson's r) between X and Y, follow these steps:

Step 1: Calculate the means (average) for both X and Y.

Mean(X) = (7 + 4 + 6 + 9 + 3 + 8) / 6 = 37 / 6 = 6.17 (rounded to 2 decimal places)

Mean(Y) = (8 + 5 + 4 + 8 + 3 + 6) / 6 = 34 / 6 = 5.67 (rounded to 2 decimal places)

Step 2: Calculate the sum of the products of the deviations of X and Y from their respective means:

Σ((X - Mean(X)) * (Y - Mean(Y))) = ((7 - 6.17) * (8 - 5.67)) + ((4 - 6.17) * (5 - 5.67)) + ((6 - 6.17) * (4 - 5.67)) + ((9 - 6.17) * (8 - 5.67)) + ((3 - 6.17) * (3 - 5.67)) + ((8 - 6.17) * (6 - 5.67))

Σ((X - Mean(X)) * (Y - Mean(Y))) = (1.715 * 2.33) + (-2.33 * -0.67) + (-0.17 * -1.67) + (2.83 * 2.33) + (-3.17 * -2.67) + (1.83 * 0.33)

Σ((X - Mean(X)) * (Y - Mean(Y))) = 3.99095 + 1.5651 + 0.2845 + 6.6117 + 8.4769 + 0.6039 = 21.53315

Step 3: Calculate the sum of the squared deviations of X and Y from their respective means.

Σ((X - Mean(X))²) = ((7 - 6.17)² + (4 - 6.17)² + (6 - 6.17)² + (9 - 6.17)² + (3 - 6.17)² + (8 - 6.17)²)

Σ((X - Mean(X))²) = (1.715² + 2.33² + 0.17² + 2.83² + 3.17² + 1.83²)

Σ((X - Mean(X))²) = 2.941225 + 5.4289 + 0.0289 + 7.9889 + 10.0489 + 3.3489 = 30.7879

Σ((Y - Mean(Y))²) = ((8 - 5.67)² + (5 - 5.67)² + (4 - 5.67)² + (8 - 5.67)² + (3 - 5.67)² + (6 - 5.67)²)

Σ((Y - Mean(Y))²) = (5.3109 + 0.4489 + 2.7889 + 5.3109 + 7.9889 + 0.1089)

Σ((Y - Mean(Y))²) = 21.5464

Step 4: Calculate Pearson's correlation coefficient using the formula:

r = Σ((X - Mean(X)) * (Y - Mean(Y))) / √(Σ((X - Mean(X))²) * Σ((Y - Mean(Y))²))

r = 21.53315 / √(30.7879 * 21.5464)

r ≈ 0.942 (rounded to three decimal places)

So, the Pearson correlation coefficient (r) is approximately 0.942, indicating a strong positive linear correlation between X and Y.