Surveys and samples

A survey asks a sample – some of the people – to learn about the whole population: all the people the question is about. The percent in the sample is an estimate of the percent in the whole population.

A school has 1,200 pupils.
The pupils’ council asked 150 of them whether they want a longer break, and 60 answered “yes”.

• The percent in the sample: $\displaystyle {\frac{60}{150}}={0.4}$, that is 40%.
• The estimate for the whole school: $\displaystyle {0.4\cdot1200}={480}$ pupils.

This is an estimate, not a count: had 150 other pupils been asked, the result would have come out a little different.

The estimate is good only when the sample is representative: like the population in everything that could affect the answer. A sample that is not representative is called a biased sample.

Common ways to get a biased sample:

• Asking in one place – a survey about a new shopping centre carried out in the old one asks only people who already shop in shopping centres.
• Those who choose to answer – in an online survey or a vote by text message, mostly people who care a lot about the subject answer.
• A group easy to reach – friends in the class, followers on a social network.

The best way: choose the participants at random from the whole population, so that everyone has the same chance of being chosen.

Even with a representative sample, a badly worded question pushes the answers one way.

• “Do you support a bigger budget for public parks, which every family enjoys?” – the question hints at the right answer.
• “Do you support a bigger budget for public parks?” – a neutral question.

A survey’s result is an estimate, so a margin of error is published with it: by about how many percent the result in the population may differ from the result in the sample.

An election poll asked 500 voters whom they would vote for.
The margin of error: 4.4%.

• Party C got 12% in the poll. In the whole population its support is probably between $\displaystyle {12-4.4}={7.6}$ percent and $\displaystyle {12+4.4}={16.4}$ percent.
• The Knesset has 120 seats. 12% of them is $\displaystyle {0.12\cdot120}={14.4}$, about 14 seats.
• Party A got 23% and party B 20%. The difference, 3%, is smaller than the margin of error: the poll does not show that party A leads.
• The larger the sample, the smaller the margin of error: in a poll of 2,000 people it is about half that of a poll of 500.

A town has 40,000 residents.
The council asked 250 residents, chosen at random, whether to turn a car park into a public garden.
90 of them answered “yes”.

What percent of the sample support the garden?

Estimate: how many of the town’s residents support the garden?

Is the sample representative? If it is biased, explain why.

1. To learn what residents think of a new bus line, passengers at the central bus station were asked.
2. To learn how many pupils eat breakfast, 100 names were chosen at random from the list of all the school’s pupils, and they were asked.
3. A television programme asked its viewers to vote by text message: should phones be banned in schools?

An election poll asked 600 voters.
Party A got 25%, and party B 22%.
The margin of error: 4%.

What is the lowest percent of support for party A in the whole population, by the poll and its margin of error?

How many seats out of 120 would party A get, by the result of the poll?

A newspaper wrote: “Party A leads”. Does the poll show that? Explain.

A drinks company’s survey asked:
“Do you, like most teenagers, prefer a cold fizzy drink to water?”

What is wrong with the question? Write a neutral question instead.

Two surveys asked the same question:
Survey A – 400 people, margin of error 5%.
Survey B – 1,600 people, margin of error 2.5%.

In survey A 52% answered “yes”, and in survey B 47%.
Which survey would you trust more? Can you say that most of the population answers “yes”? Explain.