A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are:
14 students: Science Club | 11 students: Arts Club |
9 students: Sports Club | 6 students: Debate Club
Assume there are 800 students in the whole school.
- (i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
- (ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
Step-by-step solution
Idea: Relative frequency in the sample = probability for the sample. Multiply that probability by the size of the whole school to estimate the number in the school.
(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
- Total in the sample: 14 + 11 + 9 + 6 = 40. Arts Club: 11.½ mark
- P(Arts Club) = 1140 = 0.275 (27.5%).½ mark
(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
- In the sample, P(Sports Club) = 940 = 0.225.½ mark
- The sample is random, so we expect about the same share in the whole school of 800.½ mark
- Estimate = 940 × 800 = 9 × 20 = 180 students.1 mark
Check: Scale factor 800 ÷ 40 = 20: Science 280, Arts 220, Sports 180, Debate 120. Total 800 ✓.
Answer to write in the exam
(i)
Total in sample = 14 + 11 + 9 + 6 = 40; Arts Club = 11
∴ P(Arts Club) = 1140 = 0.275
(ii)
P(Sports Club) in sample = 940
Estimate = 940 × 800 = 180
∴ About 180 students are likely to prefer the Sports Club
Common mistakes that cost marks
- Using 800 as the total in (i). Part (i) is about the sample, so the total is 40.
- In (ii), writing 9 × 800 = 7200 and forgetting to divide by 40.
- Giving the answer to (ii) as 0.225. The question asks for a number of students, so multiply by 800.
How this can come in the exam
In a random sample of 50 families in a town, 18 own a car. The town has 2000 families. The estimated number of families that own a car is
- 18
- 360
- 720
- 1800
Show answer
(C) 720
1850 × 2000 = 0.36 × 2000 = 720.
In a random sample of 30 students, 12 travel by school bus, 10 walk and 8 come by car. Estimate how many of the 900 students in the school walk to school.
Show answer
P(walk) = 1030 = 13. Estimate = 13 × 900 = 300 students.Try one yourself
In a random sample of 25 students, 7 prefer the Music Club. Estimate how many of the 1000 students in the school prefer the Music Club.
Show answer
725 × 1000 = 0.28 × 1000 = 280 students.
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