CA Foundation P3 · Chapter 17

Correlation and Regression MCQs with Answers

9 multiple-choice questions on Correlation and Regression for CA Foundation P3 Quantitative Aptitude. Try each one before revealing the answer and explanation.

1. Question 1

   The coefficient of correlation r always lies between:

   • A) −1 and +1
   • B) 0 and 1
   • C) −∞ and +∞
   • D) 0 and ∞
   Show answer & explanation

   Answer: A) −1 and +1

   Karl Pearson's coefficient of correlation is a pure number satisfying −1 ≤ r ≤ +1. r = +1 means perfect positive and r = −1 perfect negative correlation.

2. Question 2

   If Cov(x, y) = 18, σx = 4 and σy = 6, the coefficient of correlation is:

   • A) 1.33
   • B) 3
   • C) 0.075
   • D) 0.75
   Show answer & explanation

   Answer: D) 0.75

   r = Cov(x, y)/(σx σy) = 18/(4 x 6) = 18/24 = 0.75. Inverting the ratio gives 1.33, which is impossible since |r| cannot exceed 1.

3. Question 3

   The two regression coefficients are byx = 0.8 and bxy = 0.45. The coefficient of correlation is:

   • A) 0.6
   • B) 0.36
   • C) 0.625
   • D) 1.25
   Show answer & explanation

   Answer: A) 0.6

   r = ±√(byx x bxy) = √(0.8 x 0.45) = √0.36 = 0.6, positive because both coefficients are positive. 0.36 is r² (coefficient of determination), not r.

4. Question 4

   For 6 pairs of ranks, the sum of squares of rank differences Σd² = 14. Spearman's rank correlation coefficient is (to two decimals):

   • A) 0.40
   • B) −0.60
   • C) 0.93
   • D) 0.60
   Show answer & explanation

   Answer: D) 0.60

   R = 1 − 6Σd²/[n(n² − 1)] = 1 − (6 x 14)/(6 x 35) = 1 − 84/210 = 1 − 0.40 = 0.60. Omitting the factor 6 in the numerator gives 1 − 14/210 = 0.93, 0.40 forgets to subtract from 1, and −0.60 has the wrong sign.

5. Question 5

   The two regression lines are 3x + 2y = 26 and 6x + y = 31. The means of x and y are:

   • A) x̄ = 7, ȳ = 4
   • B) x̄ = 3, ȳ = 8.5
   • C) x̄ = 4, ȳ = 7
   • D) x̄ = 5, ȳ = 5.5
   Show answer & explanation

   Answer: C) x̄ = 4, ȳ = 7

   Both regression lines pass through (x̄, ȳ), so solve them together. From the second, y = 31 − 6x. Substituting: 3x + 62 − 12x = 26, so 9x = 36 and x = 4; then y = 31 − 24 = 7. Hence x̄ = 4, ȳ = 7.

6. Question 6

   The two regression lines are 3x + 2y = 26 and 6x + y = 31. The coefficient of correlation between x and y is:

   • A) 0.5
   • B) −0.5
   • C) −0.25
   • D) −2
   Show answer & explanation

   Answer: B) −0.5

   Assume 3x + 2y = 26 is y on x: y = 13 − 1.5x, so byx = −1.5. Then 6x + y = 31 is x on y: x = 31/6 − y/6, so bxy = −1/6. The product 0.25 ≤ 1, so the assumption is valid (the reverse gives a product of 4, which is impossible). r = −√0.25 = −0.5, negative because both coefficients are negative.

7. Question 7

   The coefficient of correlation from 25 pairs of observations is 0.8. Taking PE = 0.6745 x (1 − r²)/√n, the probable error of r is (to four decimals):

   • A) 0.0720
   • B) 0.0486
   • C) 0.0270
   • D) 0.2428
   Show answer & explanation

   Answer: B) 0.0486

   PE = 0.6745 x (1 − r²)/√n = 0.6745 x (1 − 0.64)/5 = 0.6745 x 0.072 = 0.0486. 0.0720 is the standard error (without 0.6745), 0.0270 uses (1 − r) instead of (1 − r²), and 0.2428 forgets to divide by √n.

8. Question 8

   Given x̄ = 20, ȳ = 45 and the regression coefficient of y on x, byx = 1.2, the estimated value of y when x = 25 is:

   • A) 30
   • B) 75
   • C) 39
   • D) 51
   Show answer & explanation

   Answer: D) 51

   The regression line of y on x is y − ȳ = byx(x − x̄). So y = 45 + 1.2(25 − 20) = 45 + 6 = 51. Computing 1.2 x 25 alone gives 30, and subtracting the adjustment gives 39.

9. Question 9

   When the coefficient of correlation r = 0, the two lines of regression are:

   • A) Perpendicular to each other
   • B) Coincident
   • C) Parallel to each other
   • D) Inclined at 45° to each other
   Show answer & explanation

   Answer: A) Perpendicular to each other

   When r = 0, byx = bxy = 0, so the line of y on x is y = ȳ (horizontal) and the line of x on y is x = x̄ (vertical). These are perpendicular. When r = ±1 the two lines coincide.