11.2 Sampling techniques and their critique — Practise with help

Work each question closed book first. For a criticism, name the part of the population that the method is most likely to miss.

Warm-up: remember

Work out $20\%$ of $350$ .
Simplify $\dfrac{480}{800}$ .
Write the next number in the pattern $7, 27, 47, \ldots$

Answers

$0.2 \times 350 = 70$ .
$\dfrac{480}{800} = \dfrac{3}{5}$ (divide both parts by 160).
$67$ . The pattern adds $20$ each time, so a systematic sample with interval $k = 20$ follows it.

If question 2 was hard, revise equivalent fractions before you start.

Practise with help

1. A club has 240 members. A simple random sample of 30 is chosen. Find the probability that a given member is in the sample.

Hint

In a simple random sample, the probability is the sample size divided by the population size.

Next step

Work out $\dfrac{30}{240}$ and simplify it.

Full working

\frac{30}{240} = \frac{1}{8}.

Every member has chance $\dfrac{1}{8}$ , because every member of the frame is equally likely to be chosen.

2. A population of 900 is sampled systematically with sample size 45. Find the interval, then list the first three members if the start is chosen at random as 4.

Hint

The interval is the population size divided by the sample size.

Next step

$k = \dfrac{900}{45} = 20$ . Add $20$ twice to the starting value 4.

Full working

k = \frac{900}{45} = 20.

Starting at 4, the first three members are 4, 24 and 44.

3. A school has 600 girls and 400 boys. A stratified sample of 50 is needed. Find the number of girls and the number of boys in the sample.

Hint

The population is $600 + 400 = 1000$ . Each group contributes in proportion to its size.

Next step

For the girls, work out $\dfrac{600}{1000} \times 50$ . Then do the same for the boys.

Full working

n_{\text{girls}} = \frac{600}{1000} \times 50 = 30, \qquad n_{\text{boys}} = \frac{400}{1000} \times 50 = 20.

The two sample sizes add to 50, which is the check that the allocation is complete.

4. A researcher surveys the first 60 people who leave a gym between 6 pm and 7 pm. Name the sampling method and give one reason the sample may be biased.

Hint

The researcher chose the people who were easiest to reach.

Next step

Compare the people at the gym in that hour with everyone the researcher wants to describe.

Full working

This is an opportunity sample. It is biased because it only reaches people who use that gym, at that time. People who exercise elsewhere, at other times, or not at all are missing. The sample is large, but size does not remove bias.

5. A population of 3000 is split into strata of sizes 1200, 900 and 900. A stratified sample of 100 is taken. Find each stratum sample size.

Hint

Use $n_i = \dfrac{N_i}{N} \times n$ for each stratum.

Next step

For the first stratum, work out $\dfrac{1200}{3000} \times 100 = 40$ . Repeat with 900 for the other two.

Full working

\frac{1200}{3000} \times 100 = 40, \qquad \frac{900}{3000} \times 100 = 30, \qquad \frac{900}{3000} \times 100 = 30.

The three parts add to $40 + 30 + 30 = 100$ , which matches the total sample size.

6. A researcher wants to know the mean number of books read by adults in a town. He asks people in a library. Explain why his estimate is likely to be too high, and suggest one better method.

Hint

Think about who visits a library, and who does not.

Next step

Compare library users with the whole population of adults in the town.

Full working

People in a library are more likely to read than adults in general, so the estimate of the mean is likely to be too high. A better method is a simple random sample from a complete sampling frame, such as the electoral register, or a stratified sample with strata that cover the whole town. A random method reaches readers and non-readers alike.