Unit 8
9 marks · NormalA study collected the number of books read by each member of a club last month. The numbers are shown in the table:
| Number of books (x) | Frequency (f) |
|---|---|
| 2 | 3 |
| 3 | 5 |
| 4 | 6 |
| 5 | 2 |
| 6 | 4 |
Work out the modal number of books in the table above.
Write a function such that gives the total number of books for a given number of members, where each member reads exactly books in a month. Express in terms of and .
In April, a club member read twice as many books as in March every day. Let the number of books the member read each day in March be . Write a composite function that gives the number of books read each day in April, using your function from (b) and the rule .
Write explicitly.
Suppose the probability that a member is selected at random and has read more than 4 books in March is . Work out using the frequency data from the table.
If two members are selected at random, with replacement, what is the probability that both of them read more than 4 books in March?
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 4
(b) f(x) = n x
(c) h(x) = 2 n x
(d)
(e) 0.5 \times 0.5 = 0.25
Common mistakes: Using wrong frequency for probability; Not evaluating composite function; Reversing function rule order
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · Mode correctly identified as 4.
M1 (b) · 1 mark · Sets up a general form for the function; includes a variable for number of members and books.
A1 (b) · 1 mark · Expresses f(x) = n x or equivalent.
M1 (c) · 1 mark · Forms the correct composite function, using correct function composition order.
A1 (c) · 1 mark · States h(x) = 2 n x or equivalent.
M1 (d) · 1 mark · Finds total frequency and correct 'more than 4 books' frequency.
A1 (d) · 1 mark · Calculates correct probability as 0.5 (accept 1/2).
M1 (e) · 1 mark · Identifies the need to square the single probability with replacement.
A1 (e) · 1 mark · States probability as 0.25 (accept 1/4).
Worked solution
a) The mode is the value with the highest frequency. From the table, 4 books appears 6 times, the highest frequency.
b) If each member reads x books, and there are n members, then the total books read is n multiplied by x: f(x) = n x.
c) Composite function: In April, each member reads twice as many as in March: g(x) = 2x. We want h(x) = f(g(x)) = n × (2x) = 2 n x.
d) Total frequency = 3 + 5 + 6 + 2 + 4 = 20. Number who read more than 4 books: 5 books (2) + 6 books (4) = 6. Probability: 6/20 = 0.3. Correction: The question asks for 'read more than 4 books', i.e. 5 or 6. Therefore, number is 2 + 4 = 6, so p = 0.3 (should have added 2 + 4, so final answer is 0.3). Correction to marking scheme: Only part (d): p = 0.3 (accept 3/10).
e) With replacement, probability both have read more than 4 books: p × p = 0.3 × 0.3 = 0.09.