Chi-squared faker
Problem
The $\chi^2$ test statistic is given by
$$
\chi^2 = \sum\frac{(f_o-f_e)^2}{f_e}
$$
The weights of a certain type of primate are known through extensive studies to take an expected distribution, given by the expected value below. The weights of a community of these primates from a different location are measured, and are listed in the observed values below:
| Weight (kg) | [0, 9] | [10,19] | [20,29] | [30,39] | [40,49] | [50,59] |
| Expected | 3 | 3 | 3 | 4 | 8 | 9 |
| Observed | 5 | 6 | 3 | 5 | 7 | 7 |
| Weight (kg) | [60,69] | [70,79] | [80,89] | [90,99] | [100,109] | [110,119] | [120+] |
| Expected | 11 | 12 | 8 | 10 | 4 | 12 | 13 |
| Observed | 12 | 17 | 7 | 2 | 12 | 16 | 15 |
How would you describe the expected distribution? Can you think of a good explanation for this pattern of expected data?
You are asked to undertake a Chi-squared test to assess the hypothesis that the weights of the two populations are driven by the same distribution.
Supposing that for unscientific reasons you were keen on rejecting the hypothesis. Before making any detailed calculations, what would be the best way to proceed with the Chi-squared test to make this happen?
Conversely, how might you organise your calculation to maximise the chance of accepting the hypothesis? If you can think of several ways in which to do this, which seems most natural?
Perform the tests to see if you were correct.
Do you think that the data should be accepted or rejected at the 1% significance level?
NOTES AND BACKGROUND
As Benjamin Disraeli famously said, 'There are lies, damned lies and statistics'. This problem shows that the notion of 'significance' is not necessarily as clearly cut as the layman might imagine: data can often easily be manipulated to present a variety of possibly misleading pictures. Sometimes this manipulation is purposeful and sometimes due to 'blind' application of an algorithm. Trained statisticians often reserve a sceptical eye when presented with the results of significance tests and are always aware of the assumptions going into a calculation and the implications of these.
Getting Started
To perform a Chi-squared test you need to ensure that the data are grouped into expected categories so that the expected frequencies in all of the classes are 5 or more.
How might this grouping be made to increase or decrease the final test statistic?
You will need to use tables to determine the significance of the final answer.
Student Solutions
We could explain the expected profile as the right two peaks are the adult male weights, the preceding peaks are due to adolescent males. To see better, we would need to superimpose a male weight profile on top of a female weight profile.
To make a $\chi$-squared test pass we want to group data which reduces the overall statistic. To do this we would try to group a class which is an over-estimate with a class which is an under-estimate.
To make it fail, we would want to increase the statistic. To do this we would group together classes which are over-estimates and group together classes which are under-estimates.
Obviously, we would want to group classes in ways which look reasonable!
We looked at these ways to group the data:
Teachers' Resources
Why do this problem?
Possible approach
Key questions
- Can you think of a convincing explanation for the expected distribution of weights?
- What choices are there to be made in a Chi-squared calculation?
- How would you group classes to most increase the Chi-squared statistic?