These are original Marlbridge practice questions, not past exam questions. All data sets are fictional. The rubrics are a suggested Marlbridge rubric to help you check your work; they are not an official scoring guideline. Assumptions for every question: each individual gave exactly one answer, so each is counted in exactly one category; “relative frequency” means a proportion between 0 and 1; give proportions to 3 decimal places, or percentages to 1 decimal place, unless they are exact.
Question 1 (multiple choice · foundation)
A fictional juice bar recorded the flavour chosen by each of its first 80 customers on one day.
| Flavour | Orange | Apple | Mango | Grape | Total |
|---|---|---|---|---|---|
| Number of customers | 28 | 20 | 18 | 14 | 80 |
What is the relative frequency of Mango?
- (A) 0.180
- (B) 0.225
- (C) 0.290
- (D) 0.643
Answer and explanation
Answer: (B). Relative frequency = count ÷ n = 18 ÷ 80 = 0.225. So 22.5% of these 80 customers chose Mango.
- (A) divides by 100 instead of by the total of 80 customers. That only works when n = 100.
- (C) divides by 62, the number of customers who did not choose Mango (80 − 18). That is a part-to-rest ratio, 18 : 62, not a proportion of the whole.
- (D) divides by 28, the Orange count. That compares Mango with Orange, not with all customers.
Question 2 (multiple choice · core)
A fictional city council asked 300 adults which one format they prefer for news. The relative frequencies are:
| Preferred format | Printed newspaper | News website | Social media | Radio or TV |
|---|---|---|---|---|
| Relative frequency | 0.11 | 0.31 | 0.38 | 0.20 |
Which statement is supported by the table?
- (A) A majority of these adults prefer social media.
- (B) More than half of these adults prefer either a news website or social media.
- (C) Exactly 38 of these adults prefer social media.
- (D) Fewer than 1 in 10 of these adults prefer a printed newspaper.
Answer and explanation
Answer: (B). The two categories do not overlap, so add: 0.31 + 0.38 = 0.69. Since 0.69 > 0.5, more than half of these 300 adults prefer one of the two online formats.
- (A) confuses “most common” with “majority”. Social media is the most common choice, but 0.38 < 0.5.
- (C) treats the proportion as a count. The number is 0.38 × 300 = 114 adults.
- (D) is false: 1 in 10 is 0.10, and 0.11 > 0.10. Slightly more than 1 in 10 prefer a printed newspaper (0.11 × 300 = 33 adults).
Question 3 (multiple choice · core)
At a fictional orchard, every apple picked on Monday was either accepted or rejected. The ratio of rejected apples to accepted apples was 3 : 17. What proportion of the apples picked on Monday were rejected?
- (A) 0.030
- (B) 0.150
- (C) 0.176
- (D) 0.850
Answer and explanation
Answer: (B). A ratio of 3 : 17 means 3 rejected for every 3 + 17 = 20 apples. The proportion rejected is 3 ÷ 20 = 0.150, or 15%. The ratio and the proportion carry the same information.
- (A) reads the “3” as a percentage (3%). The ratio says nothing about 100 apples.
- (C) is 3 ÷ 17 = 0.176. That divides by the accepted apples only, not by all apples.
- (D) is 17 ÷ 20, the proportion accepted.
Question 4 (calculation · core)
A fictional café recorded the size of 20 coffee orders in the order they were made:
Large, Large, Medium, Large, Medium, Medium, Large, Small, Medium, Small, Medium, Small, Large, Medium, Medium, Small, Small, Medium, Small, Medium
(a) Build a frequency table and a relative frequency table for order size. (b) What percentage of these orders were not Small? Interpret your answer.
Worked solution
(a) Count each size once, working through the list in order:
| Size | Frequency | Relative frequency |
|---|---|---|
| Small | 6 | 6 ÷ 20 = 0.30 |
| Medium | 9 | 9 ÷ 20 = 0.45 |
| Large | 5 | 5 ÷ 20 = 0.25 |
| Total | 20 | 1.00 |
Check: 6 + 9 + 5 = 20 and 0.30 + 0.45 + 0.25 = 1.00. The sizes have a natural order, so the table keeps Small, Medium, Large.
(b) 1 − 0.30 = 0.70, so 70%. (Check with counts: 9 + 5 = 14, and 14 ÷ 20 = 0.70.)
Interpretation. 70% of these 20 coffee orders at the café were Medium or Large.
Suggested mark points (3): 1 for correct counts that add to 20; 1 for correct relative frequencies that add to 1; 1 for 70% with a context sentence naming the 20 orders. Dividing each count by 100 (giving 0.06, 0.09, 0.05) does not earn the second point.
Question 5 (constructed response · core)
A fictional bookshop asked 200 customers to name their favourite type of book. The manager’s table has one value missing.
| Favourite type | Fiction | Mystery | Science fiction | Biography | Other |
|---|---|---|---|---|---|
| Relative frequency | 0.35 | 0.23 | ? | 0.11 | 0.12 |
(a) Find the missing relative frequency. (b) How many of the 200 customers chose Mystery? (c) The manager says: “The majority of our customers prefer Fiction.” Is this claim supported? Explain. (d) An assistant says: “Fewer than 1 in 4 of these customers chose Mystery.” Is this claim supported? Explain.
Worked solution and suggested Marlbridge rubric
(a) The relative frequencies must add to 1: 1 − (0.35 + 0.23 + 0.11 + 0.12) = 1 − 0.81 = 0.19.
(b) 0.23 × 200 = 46 customers.
(c) Not supported. Fiction is the most common favourite (0.35, the largest relative frequency), but a majority needs more than half. Since 0.35 < 0.5, only 35% of these 200 customers (70 people) chose Fiction, and 65% chose something else.
(d) Supported. 1 in 4 is 0.25, and 0.23 < 0.25. So 23% of these customers (46 of 200) chose Mystery, which is fewer than 1 in 4.
| Point | What earns it |
|---|---|
| 1 | Missing value 0.19, using the fact that relative frequencies add to 1 |
| 1 | 46 customers, from 0.23 × 200 |
| 1 | Rejects the majority claim by comparing 0.35 with 0.5, and notes Fiction is still the most common choice |
| 1 | Supports claim (d) by comparing 0.23 with 0.25, in context |
Do not award point 3 for “Fiction is only 35%” without comparing it with one half.
Question 6 (constructed response · stretch)
A fictional sports centre asked its 120 members which one activity they use most. Every member answered.
| Activity | Gym | Swimming | Classes | Racket sports | Other | Total |
|---|---|---|---|---|---|---|
| Number of members | 45 | 30 | 24 | 15 | 6 | 120 |
(a) Build a relative frequency table, giving each value as a proportion and as a percentage. (b) The manager writes: “Gym users outnumber swimmers by 3 to 2.” Show that this ratio gives the same information as the counts and as the proportions. (c) A board member thinks the form let members tick every activity they use. Use the table to explain why this is unlikely. What would you expect to see in the table if many members had ticked two activities?
Worked solution and suggested Marlbridge rubric
(a)
| Activity | Proportion | Percentage |
|---|---|---|
| Gym | 45 ÷ 120 = 0.375 | 37.5% |
| Swimming | 30 ÷ 120 = 0.250 | 25.0% |
| Classes | 24 ÷ 120 = 0.200 | 20.0% |
| Racket sports | 15 ÷ 120 = 0.125 | 12.5% |
| Other | 6 ÷ 120 = 0.050 | 5.0% |
| Total | 1.000 | 100% |
(b) Counts: 45 : 30. Dividing both by 15 gives 3 : 2. Proportions: 0.375 : 0.250, and 0.375 ÷ 0.250 = 1.5 = 3 ÷ 2, so again 3 : 2. For every 2 members who mainly swim, 3 mainly use the gym. Dividing every count by the same n = 120 does not change the ratio, so all three forms carry the same information.
(c) The counts add to exactly 120, the number of members, and the proportions add to exactly 1. Since every member answered, this fits each member being in exactly one category. If many members had ticked two activities, the counts would add to more than 120 (for example, 40 members ticking two would give 160 ticks), and the “proportions” would add to more than 1 (160 ÷ 120 ≈ 1.333). That table would not show the distribution of one categorical variable.
| Point | What earns it |
|---|---|
| 1 | All five proportions (or percentages) correct, with a total of 1 (or 100%) |
| 1 | Shows 45 : 30 = 3 : 2 and 0.375 : 0.250 = 3 : 2 |
| 1 | Uses the total of 120 (or the sum of 1) to argue each member is counted once |
| 1 | States that multiple ticks would make the counts add to more than 120 (proportions more than 1) |
For point 2, a correct sentence such as “for every 2 swimmers there are 3 gym users” is a good addition but is not required.
Question 7 (explanation · stretch)
A fictional newsletter reports the results of a residents’ vote on a new park layout. It gives only percentages: Yes 62.5%, No 25%, Unsure 12.5%. Each resident who voted chose one option, and the percentages are exact (not rounded).
(a) What is the smallest number of residents who could have voted? Show how you know. (b) The newsletter later says that 400 residents voted. How many voted Yes, No and Unsure? (c) Explain why a reader should want to know the number of voters before accepting the claim “Yes is the clear choice of residents”.
Model answer and suggested Marlbridge rubric
(a) As fractions: 62.5% = 5/8, 25% = 2/8 and 12.5% = 1/8. Every count must be a whole number, so n must be a multiple of 8. The smallest possible number of voters is 8: 5 Yes, 2 No and 1 Unsure.
(b) Yes: 0.625 × 400 = 250. No: 0.25 × 400 = 100. Unsure: 0.125 × 400 = 50. Check: 250 + 100 + 50 = 400.
(c) The same percentages fit 8 voters or 400 voters, so the table alone hides how many people the result is based on. With 8 voters, a single resident changing from Yes to No would change Yes from 62.5% (5 of 8) to 50% (4 of 8), and Yes would no longer be a majority. With 400 voters, one person changing their vote hardly moves the percentages. So the claim is much more convincing when the reader knows that n is large.
| Point | What earns it |
|---|---|
| 1 | n = 8 with the fractions 5/8, 2/8, 1/8 (or equivalent whole-number reasoning) |
| 1 | Counts 250, 100 and 50 |
| 1 | Explains that relative frequencies alone do not show n, with a concrete illustration of how a small n makes the result fragile |
How did you do?
- Q1 or Q4 wrong: re-read “Relative frequency tables” in the study guide. Divide by n, not by 100, and check the total.
- Q2 or Q5 wrong: revisit “Using counts and relative frequencies to justify claims” and Worked example 2. Most common is not a majority.
- Q3 or Q6(b) wrong: revisit “One piece of information, four forms” and the note on part-to-whole and part-to-rest ratios.
- Q6(c) wrong: revisit “Rounding and totals that are not exactly 1”.
- Q7 wrong: work through Worked example 2(a) again, then re-read the misconception about n.
Then tick off the topic checklist.