Goodness of Fit

Goodness of fit is a measure of how well real-life observed data fits a theoretical model
- For example, modelling a coin as fair then flipping it 20 times
  - You may observe 13 heads
  - You would expect 10 heads
Observed ( $O_{i}$ ) and expected ( $E_{i}$ ) values can be shown in a table
- For example, rolling a fair die 60 times ( $N = 60$ )
  - Outcome 1 2 3 4 5 6
    
    $O_{i}$ 12 7 8 10 14 9
    
    $E_{i}$ 10 10 10 10 10 10
  - Note that $\sum_{}^{} O_{i} = \sum_{}^{} E_{i} = N = 60$
How different do observed and expected values need to be before the model is not a good fit?
- You can do a hypothesis test to reach a conclusion

$H_{0} :$ There is no difference between the observed and the expected distribution
$H_{1} :$ The observed distribution cannot be modelled by the expected distribution
Let $α %$ be the significance level

First, combine any columns for which expected values are less than 5 until they are greater than 5
- For example
  - Score 1 2 3 4
    
    $O_{i}$ 15 6 4 1
    
    $E_{i}$ 12 8 4 2
  - The expected value of 2 is less than 5 so combine the last two columns
  - Score 1 2 3+
    
    $O_{i}$ 15 6 5
    
    $E_{i}$ 12 8 6
Then calculate the goodness of fit, $X^{2}$ , from the formula
- $X^{2} = \sum_{}^{} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}$
An alternative version of the formula that can be easier to calculate is
- $X^{2} = \sum_{}^{} \frac{{O_{i}}^{2}}{E_{i}} - N$
  - Where $N$ is the sum of all observed values
  - $N = \sum_{}^{} O_{i}$
  - This is also the same as the sum of all expected values
The larger $X^{2}$ is, the more different the observed values are from the expected values

The number of degrees of freedom, $ν$ , is equal to
- The number of columns (after combining to get $E_{i} > 5$ ) subtract 1
If you also use the observed data to estimate a parameter, then you subtract 2 instead
- For example, trying to estimate $p$ when comparing to a $B (n, p)$ distribution
You are subtracting the number of constraints (or restrictions)
- This is the number of times you use the observed data to help form the expected data
  - This is always 1 from ensuring their totals match, $\sum_{}^{} E_{i} = \sum_{}^{} O_{i}$
  - Then another 1 for each parameter estimated

Once you have calculated the goodness of fit, $X^{2}$
- Compare it to the critical value $χ_{ν}^{2}$ from the chi-squared distribution
  - $ν$ is the number of degrees of freedom
  - Tables of critical values are provided in the exam
  - You need the significance level, $α %$
  - All chi-squared tests are one-tailed
- If $X^{2} < χ_{ν}^{2} (α %)$ then there is insufficient evidence to reject $H_{0}$
  - This means there is no difference between the observed and expected distributions
  - In other words, "the expected distribution is a suitable model for the data"
- If $X^{2} > χ_{ν}^{2} (α %)$ then there is sufficient evidence to reject $H_{0}$
  - The expected distribution is not a suitable model for the data
Alternatively, you can use your calculator to find the $χ_{ν}^{2}$ p-value
- This is the probability of obtaining a chi-squared value of $X^{2}$ or more
- If $p < α$ then the result is critical (reject $H_{0}$ )

The alternative formula $X^{2} = \sum_{}^{} \frac{{O_{i}}^{2}}{E_{i}} - N$ is not given in the Formulae Booklet

A game is meant to award points according to the probability distribution below.

Points	2	4	8	10
Probability	0.6	0.2	0.15	0.05

The game is played by 40 people, giving the results below.

Points	2	4	8	10
Frequency	28	5	4	3

Test, at the 5% level of significance, whether or not the game is operating correctly.

goodness-of-fit-1 goodness-of-fit-2

Goodness of Fit (Edexcel A Level Further Maths: Further Statistics 1)