# Investigation: Obtaining representative data

Lesson

Did you know that by analyzing a small sample that is representative of a population we can get a good idea of what the entire population is like? You have probably had experience with this before, whether knowingly or unknowingly. Consider the following scenario:

You have just finished baking $96$96 cookies. Before you take them to your friend’s party, you have to check whether they taste good or not, so you decide to do a taste test. How many cookies should you sample?

How many cookies should you sample to check that they taste good?

All of them? Of course not! If you ate all of them, there wouldn’t be any left for your friend’s party! Instead, you'll have to rely on a random sample of the cookies.

#### Discussion questions

##### Question 1

Rob, Nick and Marcel are having an argument about what the most popular sport among students at their school is. Rob thinks it is football, Nick thinks it is basketball, while Marcel thinks it is baseball. To settle the argument once and for all, they decide to carry out a survey. But instead of collecting the data together as a group, they each go out on their own. Rob goes and asks his teammates in the school football team, Nick asks members in the girls dancing class and Marcel asks everyone who is stuck in after-school detention with him. The following table shows the results of their investigation.

% football % basketball % baseball
Rob 90 5 5
Nick 0 80 20
Marcel 30 30 40
1. How has the choice of sample affected the reliability of the results for each student?
2. Rob points out that the $90%$90% support for football in his survey is higher than the $80%$80% support for basketball and the $40%$40% support for baseball found in the others’ surveys, and claims that this is proof of football being more popular than the other sports. Is this claim valid? Explain your reasoning.
3. Marcel points out that the sample used in his survey is less biased than the samples used by the other two and argues that his findings are therefore more reliable. Is he right? Explain your reasoning.
4. Would your answer to the previous question change if it turned out that the majority of students in detention are there as a result of shattering windows during games of baseball during lunchtimes?
5. Explain how the students could have collected the results for the survey in a way that would be representative of the entire school.
##### Question 2

Britney is in charge of coming up with the town’s budget. One thing she is undecided on is whether to spend money on a new music hall for the town’s residents. So she decides to carry out a survey to gage the residents’ support for building a new music hall. She is also unsure how to select the sample for this survey and so consults her husband who suggests the following methods.

For each method, comment on whether the sample gathered would be representative of the town population and suggest changes that could be made to improve the method.

1. Interview students and teachers at the local music school
2. Interview passengers waiting at a bus stop and passengers alighting from the bus
3. Perform a letterbox drop of questionnaires along the street on which the music hall will be built

### Outcomes

#### 7.SP.A.1

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences

#### 7.SP.A.2

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.