11.2 Sampling techniques and their critique — Practise

Work through these on your own. For each sampling method, say whether it is random, and name any part of the population it may miss.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A school has 1200 students. A simple random sample of 60 is chosen. Find the probability that a given student is chosen.
A population of 500 is sampled systematically. The sample size is 25. Find the sampling interval $k$ .
A college has 480 Year 12 students and 320 Year 13 students. A stratified sample of 40 is taken. Find the number from Year 12.
A college has 480 Year 12 students and 320 Year 13 students. A stratified sample of 40 is taken. Find the number from Year 13.
A town has three age groups of sizes 2000, 3000 and 5000. A stratified sample of 200 is taken. Find the number from the largest group.
A population splits into strata of size 300 and 700. A stratified sample of 60 is taken. Find the number from the 700 stratum.
A factory has 2500 workers, and a sample of 125 is taken. Find the fraction of the population that is sampled.
An interviewer is told to fill a quota of 50 people, made up of 30 women and 20 men. Find the proportion of the quota that is women.
A population numbered 1 to 200 is sampled systematically. The sample size is 40 and the starting point is 7. Find the third member of the sample.
A list of 720 names is sampled systematically, and the sample size is 90. Find the interval $k$ .
A list of 1000 names is sampled by taking every 10th name. How many names are in the sample?
A population of 350 is split into strata of size 120, 150 and 80. A stratified sample of 50 is taken. Find the stratum sample size for the 120 stratum, rounded to the nearest whole person.

Answers and concise solutions

1. The probability

Answer: In a simple random sample every student has the same chance: $\dfrac{60}{1200} = \dfrac{1}{20} = 0.05$ .

2. The interval

Answer: Divide the population size by the sample size: $k = \dfrac{500}{25} = 20$ . Every 20th member is taken.

3. The Year 12 sample size

Answer: The population is $800$ , so the Year 12 sample is $\dfrac{480}{800} \times 40 = 24$ students.

4. The Year 13 sample size

Answer: The Year 13 sample is $\dfrac{320}{800} \times 40 = 16$ students. The two sample sizes add to $40$ , as they must.

5. The sample size from the largest group

Answer: The population is $10000$ . The largest group gives $\dfrac{5000}{10000} \times 200 = 100$ people, which is half of the sample, because that group is half of the population.

6. The sample size from the 700 stratum

Answer: The population is $1000$ , so the 700 stratum gives $\dfrac{700}{1000} \times 60 = 42$ people.

7. The sampling fraction

Answer: The fraction is $\dfrac{125}{2500} = \dfrac{1}{20} = 0.05$ , so $5\%$ of the population is sampled.

8. The proportion

Answer: The proportion is $\dfrac{30}{50} = \dfrac{3}{5} = 0.6$ . A quota controls the proportions, but the interviewer still chooses by convenience.

9. The third member

Answer: The interval is $k = \dfrac{200}{40} = 5$ . Starting at 7, the members are 7, 12 and 17.

10. The interval

Answer: The interval is $k = \dfrac{720}{90} = 8$ : every 8th name is taken.

11. The sample size

Answer: Taking every 10th name from 1000 names gives $\dfrac{1000}{10} = 100$ names.

12. The stratum sample size

Answer: Proportional allocation gives $\dfrac{120}{350} \times 50 = \dfrac{6000}{350} = 17.14\ldots$ , which rounds to 17 people. State that a whole person cannot be split.