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3 partsStep-by-step solution
6 stepsPart a(i)
Finite set and infinite set;
Differentiate between each of the following types of sets.
- A finite set is a set whose elements can be counted, and the counting process eventually ends. It has a definite, limited number of elements.
- An infinite set is a set whose elements cannot be counted, as the counting process would never end. It has an unlimited number of elements.
Part a(ii)
Empty set and universal set.
- An empty set (or null set) is a set that contains no elements. It is denoted by or {}.
- A universal set is a set that contains all elements relevant to a particular context or problem. It is usually denoted by .
Part b(i)
Harmonic Mean (HM)
Calculate the harmonic mean, mode, and median for the given data.
First, we prepare a table with midpoints (), frequencies (), , , and cumulative frequencies ().
| Marks | Number of Students () | Midpoint () | Cumulative Frequency () | ||
|---|---|---|---|---|---|
| 0 - 10 | 2 | 5 | 0.2 | 0.4 | 2 |
| 10 - 20 | 7 | 15 | 0.066667 | 0.466669 | 9 |
| 20 - 30 | 15 | 25 | 0.04 | 0.6 | 24 |
| 30 - 40 | 25 | 35 | 0.028571 | 0.714275 | 49 |
| 40 - 50 | 45 | 45 | 0.022222 | 1.0 | 94 |
| 50 - 60 | 32 | 55 | 0.018182 | 0.581824 | 126 |
| 60 - 70 | 15 | 65 | 0.015385 | 0.230775 | 141 |
| 70 - 80 | 9 | 75 | 0.013333 | 0.12 | 150 |
| Total |
The formula for the harmonic mean for grouped data is:
Sum the frequencies () and the values of .
Calculate the harmonic mean.
The harmonic mean is .
Part b(ii)
Mode
The formula for the mode for grouped data is:
The modal class is 40 - 50 (highest frequency, ).
(lower boundary of the modal class)
(frequency of the modal class)
(frequency of the preceding class, 30 - 40)
(frequency of the succeeding class, 50 - 60)
(class width)
Substitute the values into the formula.
Simplify the expression.
The mode is .
Part b(iii)
Median
The formula for the median for grouped data is:
First, find the median position: .
The median class is 40 - 50 (where the 75th observation falls, as goes from 49 to 94).
(lower boundary of the median class)
(total frequency)
(cumulative frequency of the class before the median class)
(frequency of the median class)
(class width)
Substitute the values into the formula.
Simplify the expression.
The median is .
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