(a) The universal set U is the set of integers, P, Q and R are subsets of U defined as follows:
\(P = x : x \leq 2 \) ; \(Q = x : -7 < x < 15\) ; \(R = x : -2 \leq x < 19\).
Find (i) \(P \cap Q\) ; (ii) \(P \cap (Q \cup R’)\), where R’ is the complement of R with respect to U.
(b) The following data shows the marks of 40 students in a History examination.
41 52 37 56 63 48 65 46 54 32 51 66 74 23 35 61 58 44 49 53 45 57 56 38 59 28 50 49 67 56 36 45 79 68 43 56 26 47 55 71.
(i) Form a grouped frequency table with the class intervals 20 – 29, 30 – 39, 40 – 49 etc; (ii) Find the mean of the distribution.
Explanation
(a) \(P = x : x \leq 2 = {..., -1, 0, 1, 2}\)
\(Q = x : -7 < x < 15 = {-6, -5, -4, -3, -2, -1, 0, ..., 13, 14}\)
\(R = x : -2 \leq x < 19 = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}\)
(i) \(P \cap Q = {-6, -5, -4, -3, -2, -1, 0, 1, 2}\)
(ii) \(R' = {..., -5, -4, -3, 19, 20, ...}\)
\(Q \cup R' = {..., -1, 0, 1, 2, 3, ...}\)
\(\therefore P \cap (Q \cup R') = {..., -1, 0, 1, 2}\)
(b)(i)
| Class intervals | Midpoint (x) | Freq (f) | \(fx\) |
| 20 - 29 | 24.5 | 3 | 73.5 |
| 30 - 39 | 34.5 | 5 | 172.5 |
| 40 - 49 | 44.5 | 10 | 445 |
| 50 - 59 | 54.5 | 13 | 708.5 |
| 60 - 69 | 64.5 | 6 | 387 |
| 70 - 79 | 74.5 | 3 | 223.5 |
| \(\sum \) | 40 | 2010 |
(ii) \(Mean = \frac{\sum fx}{\sum f} = \frac{2010}{40}\)
= \(50.25\)