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WAEC 2009 · Paper 2 · Q15

In a test, eight students obtained the following marks in Biology and Physics.

Student A B C D E F G H
Biology 76 52 63 48 84 36 28 70
Physics 75 78 28 45 56 71 54 58
  1. (a)

    (i) Calculate, correct to two decimal places, Spearman's rank correlation coefficient of the distribution. (ii) Comment on your result in (a)(i).

  2. (b)(i)

    Three-digit numbers are to be formed from 1, 2, 3, 4 and 5. If repetition is not allowed, how many numbers can be formed?

  3. (b)(ii)

    What is the probability of selecting an odd number from the numbers formed in (b)(i)?

Worked solution (try it first)

(a)(i)

  1. Rank the Biology marks (highest = 1): A 2, B 5, C 4, D 6, E 1, F 7, G 8, H 3.
  2. Rank the Physics marks: A 2, B 1, C 8, D 7, E 5, F 3, G 6, H 4.
  3. Differences dd: 0,4,−4,−1,−4,4,2,−10, 4, -4, -1, -4, 4, 2, -1, so ∑d2=0+16+16+1+16+16+4+1\sum d^2 = 0 + 16 + 16 + 1 + 16 + 16 + 4 + 1
    =70= 70.
  4. ρ=1−6∑d2n(n2−1)\rho = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}
    =1−6×708×63= 1 - \dfrac{6 \times 70}{8 \times 63}
    =1−420504= 1 - \dfrac{420}{504}
    =0.17= 0.17 (2 d.p.).

(ii)

  1. ρ\rho is positive but close to 0, so there is only a weak positive correlation between the marks in the two subjects.

(b)(i)

  1. There are 5 choices for the first digit, 4 for the second and 3 for the third: 5×4×3=605 \times 4 \times 3 = 60 numbers.

(ii)

  1. An odd number ends in 1, 3 or 5: 3 choices for the last digit, then 4 and 3 for the others, giving 3×4×3=363 \times 4 \times 3 = 36.
  2. Probability =3660=35= \dfrac{36}{60} = \dfrac35.

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