Statistics 6 - Chi-squared tests

Question 1

What is goodness of fit?

Answer 1

Goodness of fit is concerned with measuring how well an observed frequency distribution fits a known distribution.
The null hypothesis, H₀, is that there is no difference between the observed and theoretical distribution.
The alternative hypothesis, H₁, is that the is a difference between the observed and theoretical distribution.
In order to tell how closely the model fits the observed results, a measure of the goodness of fit between the observed frequencies and the expected frequencies is needed.
The measure of goodness of fit is:
𝑋²=Σ(ⁿᵢ₌₁)((𝑂ᵢ-𝐸ᵢ)²/𝐸ᵢ)
Where:
𝑂ᵢ=an observed frequency
𝐸ᵢ=an expected (theoretical) frequency, asserted by the null hypothesis
An easier to use version of the formula is:
𝑋²=Σ(ⁿᵢ₌₁)(𝑂ᵢ²/𝐸ᵢ)-𝑁
Where:
𝑂ᵢ=an observed frequency
𝐸ᵢ=an expected (theoretical) frequency, asserted by the null hypothesis
𝑁=Σ𝐸=Σ𝑂