Question 19 of 19intermediate🔧 ApplyNumerical4 marks

Calculate the correlation coefficient between X and Y and comment on their relationship X: 1, 3, 4, 5, 7, 8 Y: 2, 6, 8, 10, 14, 16 (Ans. r = 1)

Correct Answer

Step 1: Construct the table and calculate required values

XYX²Y²XY
12142
3693618
48166432
5102510050
7144919698
81664256128
∑X=28∑Y=56∑X²=164∑Y²=656∑XY=328

N = 6

Step 2: Apply the correlation coefficient formula

r = [∑XY - (∑X)(∑Y)/N] / [∑X2−(∑X\sqrt{∑X² - (∑X}²/N) × ∑Y2−(∑Y\sqrt{∑Y² - (∑Y}²/N)]

Numerator = 328 - (28×5628 \times 56)/6 = 328 - 261.33 = 66.67

Denominator = 164−784/6\sqrt{164 - 784/6} × 656−3136/6\sqrt{656 - 3136/6} = 164−130.67\sqrt{164 - 130.67} × 656−522.67\sqrt{656 - 522.67} = 33\sqrt{33}.33 × 133\sqrt{133}.33 = 5.77×11.555.77 \times 11.55 = 66.67

Step 3: Calculate r

r = 66.67/66.67 = 1

Step 4: Comment on the relationship

Since r = 1, there is a perfect positive correlation between X and Y. There exists an exact linear relation between the two variables. The value of r lies between minus one and plus one, and r = 1 indicates perfect correlation. Both X and Y move in the same direction.

Exercise: EXERCISES | Q: 20 | (Page 16)
For More Understanding

Explanation

This question tests the calculation of Karl Pearson's correlation coefficient. The data shows Y = 2X for all values, indicating a perfect linear relationship. The textbook context confirms that 'If r = 1 or r = -1 the correlation is perfect and there is exact linear relation.' The step deviation method or direct method both yield r = 1, as the magnitude of r is unaffected by change of origin and scale.

Solution Steps

  1. Step 1: Calculate ∑X=28, ∑Y=56, ∑X²=164, ∑Y²=656, ∑XY=328 with N=6

  2. Step 2: Calculate numerator = ∑XY - (∑X)(∑Y)/N = 328 - 261.33 = 66.67

  3. Step 3: Calculate denominator = ∑X2−(∑X\sqrt{∑X² - (∑X}²/N) × ∑Y2−(∑Y\sqrt{∑Y² - (∑Y}²/N) = 33\sqrt{33}.33 × 133\sqrt{133}.33 = 66.67

  4. Step 4: r = 66.67/66.67 = 1, indicating perfect positive correlation with exact linear relation

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