Goodness of Fit Test (DP IB Applications & Interpretation (AI): SL): Revision Note

Chi-Squared GOF: Uniform

What is a chi-squared goodness of fit test for a given distribution?

• A chi-squared (${\chi }^2$) goodness of fit test is used to test data from a sample which suggests that the population has a given distribution

• This could be that: 

  • the proportions of the population for different categories follows a given ratio 

  • the population follows a uniform distribution

    • This means all outcomes are equally likely

What are the steps for a chi-squared goodness of fit test for a given distribution?

• STEP 1
  Write the hypotheses

  • H₀ : Variable X can be modelled by the given distribution

  • H₁ : Variable X cannot be modelled by the given distribution

    • Make sure you clearly write what the variable is and don’t just call it X

• STEP 2
  Calculate the expected frequencies

  • Split the total frequency using the given ratio

  • For a uniform distribution: divide the total frequency N by the number of possible outcomes k

• STEP 3
  Calculate the degrees of freedom for the test

  • For k possible outcomes

  • Degrees of freedom is $\nu =k-1$

• STEP 4
  Enter the frequencies and the degrees of freedom into your GDC

  • Enter the observed and expected frequencies as two separate lists

  • Your GDC will then give you the χ² statistic and its p-value

  • The χ² statistic is denoted as ${\chi }_{calc}^2$

• STEP 5
  Decide whether there is evidence to reject the null hypothesis

  • EITHER compare the χ² statistic with the given critical value

    • If χ² statistic > critical value then reject H₀

    • If χ² statistic < critical value then accept H₀

  • OR compare the p-value with the given significance level

    • If p-value < significance level then reject H₀

    • If p-value > significance level then accept H₀

• STEP 6
  Write your conclusion

  • If you reject H₀

    • There is sufficient evidence to suggest that variable X does not follow the given distribution

    • Therefore this suggests that the data is not distributed as claimed

  •  If you accept H₀

    • There is insufficient evidence to suggest that variable X does not follow the given distribution

    • Therefore this suggests that the data is distributed as claimed

Chi-Squared GOF: Binomial

What is a chi-squared goodness of fit test for a binomial distribution?

• A chi-squared (${\chi }^2$) goodness of fit test is used to test data from a sample suggesting that the population has a binomial distribution

  • You will be given the value of p for the binomial distribution

What are the steps for a chi-squared goodness of fit test for a binomial distribution?

• STEP 1
  Write the hypotheses

  • H₀ : Variable X can be modelled by the binomial distribution $\mathrm{B}(n,p)$

  • H₁ : Variable X cannot be modelled by the binomial distribution $\mathrm{B}(n,p)$

    • Make sure you clearly write what the variable is and don’t just call it X

    • State the values of n and p clearly

• STEP 2
  Calculate the expected frequencies

  • Find the probability of each outcome using the binomial distribution $\mathrm{P}(X=x)$

  • Multiply the probability by the total frequency $\mathrm{P}(X=x)\times N$

• STEP 3
  Calculate the degrees of freedom for the test

  • For k outcomes,

    • Degrees of freedom is $\nu =k-1$

• STEP 4
  Enter the frequencies and the degrees of freedom into your GDC

  • Enter the observed and expected frequencies as two separate lists

  • Your GDC will then give you the χ² statistic and its p-value

  • The χ² statistic is denoted as ${\chi }_{calc}^2$

• STEP 5
  Decide whether there is evidence to reject the null hypothesis

  • EITHER compare the χ² statistic with the given critical value

    • If χ² statistic > critical value then reject H₀

    • If χ² statistic < critical value then accept H₀

  • OR compare the p-value with the given significance level

    • If p-value < significance level then reject H₀

    • If p-value > significance level then accept H₀

• STEP 6
  Write your conclusion

  • If you reject H₀

    • There is sufficient evidence to suggest that variable X does not follow the binomial distribution $\mathrm{B}(n,p)$

    • Therefore this suggests that the data does not follow $\mathrm{B}(n,p)$

  • If you accept H₀

    • There is insufficient evidence to suggest that variable X does not follow the binomial distribution $\mathrm{B}(n,p)$

    • Therefore this suggests that the data follows $\mathrm{B}(n,p)$

Chi-Squared GOF: Normal

What is a chi-squared goodness of fit test for a normal distribution?

• A chi-squared (${\chi }^2$) goodness of fit test is used to test data from a sample suggesting that the population has a normal distribution

  • You will be given the value of μ and σ for the normal distribution

What are the steps for a chi-squared goodness of fit test for a normal distribution?

• STEP 1
  Write the hypotheses

  • H₀ : Variable X can be modelled by the normal distribution $\mathrm{N}(\mu ,{\sigma }^2)$

  • H₁ : Variable X cannot be modelled by the normal distribution $\mathrm{N}(\mu ,{\sigma }^2)$

    •  Make sure you clearly write what the variable is and don’t just call it X

    • State the values of μ and σ clearly

• STEP 2
  Calculate the expected frequencies

  • Find the probability of each outcome using the normal distribution $\mathrm{P}(a<X<b)$

    • Beware of unbounded inequalities $\mathrm{P}(X<b)$ or $\mathrm{P}(X>a)$ for the class intervals on the 'ends'

  • Multiply the probability by the total frequency $\mathrm{P}(a<X<b)\times N$

• STEP 3
  Calculate the degrees of freedom for the test

  •  For k class intervals,

    • Degrees of freedom is $\nu =k-1$

•  STEP 4
  Enter the frequencies and the degrees of freedom into your GDC

  • Enter the observed and expected frequencies as two separate lists

  • Your GDC will then give you the χ² statistic and its p-value

  • The χ² statistic is denoted as ${\chi }_{calc}^2$

• STEP 5
  Decide whether there is evidence to reject the null hypothesis

  • EITHER compare the χ² statistic with the given critical value

    • If χ² statistic > critical value then reject H₀

    • If χ² statistic < critical value then accept H₀

  • OR compare the p-value with the given significance level

    • If p-value < significance level then reject H₀

    • If p-value > significance level then accept H₀

•  STEP 6
  Write your conclusion

  •  If you reject H₀

    • There is sufficient evidence to suggest that variable X does not follow the normal distribution $\mathrm{N}(\mu ,{\sigma }^2)$

    • Therefore this suggests that the data does not follow $\mathrm{N}(\mu ,{\sigma }^2)$

  • If you accept H₀

    •  There is insufficient evidence to suggest that variable X does not follow the normal distribution $\mathrm{N}(\mu ,{\sigma }^2)$

    •  Therefore this suggests that the data follows $\mathrm{N}(\mu ,{\sigma }^2)$