Topics
9. Descriptive Statistics
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United States of AmericaFL
Grade 912

9.01 Populations and sampling

Lesson
Practice
Lesson

Concept summary

A sample survey is a research method used to collect information from a group of individuals. This information, called data, is then used to make conclusions about the population.

Population

The entire set of cases or individuals under consideration in a statistical analysis.

Sample

A set of individuals that data is collected on in a statistical experiment; a subset of the population

When the sample is representative of the population, the statistics describing the sample mean can be extended to estimate the population mean and the proportions from the sample can be used to estimate the population percentage or the size of specific groups in the population by using the population total.

Population mean

The average value, over the entire population, of a certain characteristic being measured

Population total

The total count of all cases or individuals in the population

When analyzing data, we are sometimes given a margin of error which represents a range for how far the results may differ from the actual value.

Worked examples

Example 1

For a statistical survey, the population is deemed to be all people in a city who play in any organized sporting competition. State whether the following could be a sample from this population:

a

All students in a school in the city who compete in a school sporting competition.

Approach

To determine if a sample is representative of a population, consider whether there are any people in the sample who do not fit in the described population.

Solution

Yes. The students in the school in the city are in the city and since they are participating in the school sporting competition they are participating in an organized sporting competition.

b

A random 100 people chosen from a park in the city.

Approach

To determine if a sample is representative of a population, consider whether there are any people in the sample who do not fit in the described population.

Solution

No. Although people in a city park are in the city, there is no guarantee that they also play in an organized sporting competition so this sample does not fit in the described population.

Example 2

A local council wanted to monitor the number of rabbits in the area. They used the capture-recapture technique to estimate the population of rabbits. 219 rabbits were caught, tagged, and released. Later, 42 rabbits were caught at random. 15 of these 42 rabbits had been tagged. Find the estimated population of the rabbits. Round your answer to the nearest whole number if necessary.

Approach

The sample is representative of the population and should be proportional. Identify the sample values and population values to write a proportion and solve for the missing quantity.

Solution

In this case, 15 of the 42 rabbits caught in the sample were tagged and can be used for one side of the proportional equation. Let x represent the unknown rabbit population.

1\displaystyle \frac{15}{42}\displaystyle =\displaystyle \frac{219}{x}
2\displaystyle \frac{42}{15}\displaystyle =\displaystyle \frac{x}{219}Reciprocate both fractions
3\displaystyle 613.2\displaystyle =\displaystyle xMultiply by 219

The estimated population is 613 rabbits.

Reflection

Be sure to round to a reasonable value. 613.2 rabbits does not make sense so we round this value down to 613.

Example 3

A recent survey of 250 registered adults found that only 20\% of them planned to re-elect their city's mayor with a margin of error of 3\%. If there are 5970 adults in the town, determine how many votes the current mayor can expect to receive.

Approach

The margin of error provides a range of possible values. To start the problem we need to add and subtract the margin of error from the sample statistic. Then, we can use the proportion percentages with the population total to estimate the total votes.

Solution

Using the margin of error we know that 20\% \pm 3\% of the sample, plan to vote for the current mayor. This gives a range between 17\% and 23\%. We need to convert the percentages to decimals and multiply them to the population total to find the range of votes the mayor can expect.

0.17(5970)=1014.9 \text{ and }0.23(5970)=1373.1

The mayor can expect to receive between 1014 and 1373 votes.

Outcomes

MA.912.DP.1.4

Estimate a population total, mean or percentage using data from a sample survey; develop a margin of error through the use of simulation.

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