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Samples Apples & T-tests

Thinking about samples and T-tests

In this activity you will work with different populations of apples. The goal is to take lots of samples from those populations to understand that variation occurs naturally in sampling and that the 2 sample T-test effectively tests for the likelihood of any 2 samples coming from the same population. Picking actual samples from actual populations is good concrete way of getting to this understanding

Activity

This is a very simple activity that your teacher will guide you through. The basic premise is that you will be taking lots of samples from given populations of of apples. In the first instance we are looking at variations in samples so that in then makes sense to look at how much variation in a sample can be reasonably accepted, which is the basic premise for a 2 sample T-Test.

  

You will need the cards in this spreadsheet - here also as a PDF. These are the populations of apples. There are 4 populations, Red, Green, Orange and Purple, with 500 apples in each sample. It is a more concrete experience, to print and cut out these cards, but sampling can be done from the spreadsheet too.

The task is outlined in this activity sheet - Samples, apples and T-tests. 

A ToK moment - This particular topic is a total goldmine for ToK and something I hope students can write about in their essays or exhibitions. Very specifically we are talking about trying to make valuable conclusions about entire populations from a sample. The methodology is hypothesis testing which is a neat combination of a rigorous mathematical technique often used in the human and natural sciences, leading to inductive reasoning. It is a great journey to discuss. Probabilities can be accurately calculated for very complex situations. You only need to look closely at the chi squared distribution to see that. The path from these calculations towards truth and certainty is much cloudier of course and we have to make some important decisions about the reliability of any conclusions we make form hypothesis testing. There is also the possibility to have an explicit focus on sampling and the reliability and significance of different methods. We could also zero in on the current debate about the binary nature of significance levels where two results that are super close might be either side of a arbitrary boundary that means we can draw different conclusions form them. It is all really rich material.

A global view

Teacher Notes

Cards or spreadsheets

First thing is to decide if you have time to print, laminate and cut out the cards. It takes about an hour, faster if you have a guillotine. I find that it is worth it for 2 reasons. First - it does add significantly to the concrete experience of recognising that you are taking a sample from a large population, because you actually are. Secondly, if done well, the cards are completely reusable year after year and also in different contexts and also by your colleagues in neighbouring classrooms. Perhaps you could enlist some help. I am not above asking students to help.

If you don't have the time or the means the n students can take their samples directly from the spreadsheet. They could, for example, use a random number generator to pick some rows from the spreadsheet. Up to you!

The Data

  • The Red and Green populations have exactly the same Mean, 200g and Standard deviation, 25g 
  • The Orange population has a mean of 180g and a standard deviation of 20g
  • The Purple population has a mean of 210g an a standard deviation of 30g

Students don't actually ever need to know this. In fact it is more in keeping the context for a 2 sample t-test that they don't. It is important to dwell on the assumptions involved with pooling data too. (Even if in the test we assume similar standard deviation, in the reality, it can't actually be known)

Step 1 - Prepare the populations and put them in bowls or something from which samples can easily be taken. 

Step 2 - Discuss the claim which is designed to provoke the important bit of thought. I would hope that some one observes the doubt that exists because you have taken samples. Between them and you, this is an opportunity to conclude that samples, will vary and that concluding on the strength of a sample alone is limited. We need to understand how likely that variation actually is if the samples come from the same population.

Step 3 - For part 1, the class could be split in two, with one half working with the green population and one the red. the idea is simply to experience that variation in samples is natural.

Step 4 - Then you can lead students through a 2 tailed T-test that starts with them actually taking 2 samples. One from the green and one from the red. I would stick to a 2 tailed test at this stage. Normally these tests will confirm the null hypothesis.

Step 5 - Here we can find out what happens when we do actually have different populations (Remember, they don't know this yet, its what they are trying to find out and it is a more authentic experience if the only evidence they have is a t-test)

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