The following table shows the distribution of test scores in a class.
| Scores | 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 |
| No of pupils | 1 | 1 | 5 | 3 | (k^{2} + 1) | 6 | 2 | 3 | 4 |
(a) If the mean score of the class is 6, find the : (i) value of k (ii) median score.
(b) Draw a bar chart for the distribution.
(c) If a pupil is picked at random, what is the probability that he/ she will score less than 6?
Explanation
(a)
| Scores (x) | 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | Total |
| No of pupils (f) | 1 | 1 | 5 | 3 | (k^{2} + 1) | 6 | 2 | 3 | 4 | (k^{2} + 26) |
| fx | 1 | 2 | 15 | 12 | (5k^{2} + 5) | 42 | 16 | 27 | 40 | (5k^{2} + 160) |
(bar{x} = frac{sum fx}{sum f})
(6 = frac{5k^{2} + 160}{k^{2} + 26})
(6(k^{2} + 26) = 5k^{2} + 160)
(6k^{2} + 156 = 5k^{2} + 160)
(6k^{2} – 5k^{2} = 160 – 156)
(k^{2} = 4)
( k = sqrt{4} = 2)
(therefore k = 2)
(ii) (Median = frac{5 + 7}{2})
= (frac{12}{2} = 6)
(b)
(c) Probability of scoring less than 6 = (frac{15}{30} )
= (frac{1}{2})