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 s...
Question 1 Report
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?
Answer Details
(a)(i) Finding the value of \(k\)
| Score, \(x\) | 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | Total |
|---|---|---|---|---|---|---|---|---|---|---|
| Frequency, \(f\) | 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 | \(5k^2+160\) |
Since the mean score is 6,
\[6=\frac{5k^2+160}{k^2+26}\]
\[6k^2+156=5k^2+160\]
\[k^2=4\]
\[\therefore k=2\]
Thus, the frequency for score 5 is \(k^2+1=5\), and the total number of pupils is \(4+26=30\).
(a)(ii) Median score
| Score | 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| Cumulative frequency | 1 | 2 | 7 | 10 | 15 | 21 | 23 | 26 | 30 |
There are 30 scores. The 15th score is 5 and the 16th score is 7.
\[\text{Median}=\frac{5+7}{2}=6\]
(b) Bar chart of the distribution
(c) Probability of scoring less than 6
The scores less than 6 are 1, 2, 3, 4 and 5.
\[\text{Number of such pupils}=1+1+5+3+5=15\]
\[P(\text{score less than }6)=\frac{15}{30}=\boxed{\frac{1}{2}}\]
Download The App On Google Playstore
Everything you need to excel in your exams
