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?
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.
Consider the following scenarios. Discuss your answers with a partner.
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|
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.
Imagine that you have been made in charge of investigating students’ satisfaction with the performance of your school principal. How would you go about selecting a representative sample from the entire student population? Consider who you will ask, the questions you will ask them, and where and when you will ask them.
Are the samples in the following instances representative samples? If they are not, who or what should have been included for them to have been representative?
Principal Chris is considering building a new basketball court for students to use during lunchtimes, but first, he has to find out whether students support the idea. So he decides to personally interview students to get their thoughts on the idea. But due to time constraints, he is only able to directly talk to $45$45 students.
The following table shows the number of students in each year group at a particular middle school which has $520$520 students.
|Number of students||$176$176||$180$180||$164$164|
Imagine you have just won $5$5 boxes of chocolates as part of a competition. Unfortunately, you do not eat chocolate, so you decide to give away the $5$5 boxes to students in your class. You want to be fair and so you decide to select $5$5 students at random to give the boxes of chocolates. Use a list of random numbers to select a random sample of $5$5 students from your class.
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.
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. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.