8. Probability & Statistics

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?

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 | |
---|---|---|---|

Rob | 90 | 5 | 5 |

Nick | 0 | 80 | 20 |

Marcel | 30 | 30 | 40 |

- How has the choice of sample affected the reliability of the results for each student?
- 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.
- 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.
- 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?
- Explain how the students could have collected the results for the survey in a way that would be representative of the entire school.

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.

- Interview students and teachers at the local music school
- Interview passengers waiting at a bus stop and passengers alighting from the bus
- Perform a letterbox drop of questionnaires along the street on which the music hall will be built
- Interview people entering the local newsagency, offering a $20 reward for those who agree to participate
- Post questionnaires to people whose names are longer than 20 letters on the local electoral roll

Question 3

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?

- Asking a random selection of students in your class whether they approve of the principal to find out the approval rating of the principal
- Carrying out a taste test of a large batch of cookies you have just baked by eating only a few of the burnt ones
- Conducting a survey to find the most popular teacher in the school by asking students from one class
- Carrying out a crash test of new cars just manufactured by a factory by crashing every 100th car built
- Determining the number of people in the country watching Masterchef by surveying members of a random cooking class
- Determining the average height of students in your class by measuring the height of a few randomly selected girls in your class
- Determining the number of students with blue eyes by counting the number of students with blue eyes in your class
- Determining the average heart rate of healthy students by measuring the heart rate of all students at a sports high school
- Conducting an opinion poll by surveying a random sample of AM radio listeners

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.

Grade | 6 | 7 | 8 |
---|---|---|---|

Number of students | $176$176 | $180$180 | $164$164 |

- How many students should be chosen from each year group to form a representative sample of $45$45 students?
- After talking to the sample of students, he discovers that $21$21 of them support the idea of building the new basketball court. Estimate how many students in the entire school support the idea.
- Do you think it was appropriate to use a stratified sample in this case? Explain your reasoning.

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.