Chi Squared Tests for Standard Distributions (Edexcel A Level Further Maths: Further Statistics 1): Revision Note

Chi squared for discrete uniform

How do I do a chi-squared test with a discrete uniform distribution?

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

• For a random variable $X$ with the discrete uniform distribution

  • $X$ can take a finite number $k$of distinct values

  • each value is equally likely

    • $\mathrm{P}\left(X=x\right)=\frac{1}{k},x=1,2,...,k$

• There will never be any parameters to estimate for a discrete uniform goodness of fit test

What are the steps?

• STEP 1: Write the hypotheses

  • ${\mathrm{H}}_0$ : A discrete uniform distribution is a suitable model for Variable X

  • ${\mathrm{H}}_1$ : A discrete uniform distribution is not a suitable model for Variable X 

    • The hypotheses should always be stated in the context of the question

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

• STEP 2: Calculate the expected frequencies

  • each expected frequency is the same

  • 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: Calculate $X^2$ using either version of the formula

$X^2=\sum _{i=1}^n\frac{{\left(O_i-E_i\right)}^2}{E_i}=\left(\sum _{i=1}^n\frac{{O_i}^2}{E_i}\right)-N$ 

• Determine the appropriate ${\chi }^2$ critical value

  • ${\chi }_{\nu }^2\left(\alpha \%\right)$ is the critical value with $\nu$ degrees of freedom for significance level $\alpha$ 

  • use the 'Percentage Points of the ${\chi }^2$ Distribution' table in the exam formula booklet

  • Or, alternatively, use a calculator to find the ${\chi }_{\nu }^2$ p-value

    • This is the probability of obtaining a chi-squared value of $X^2$ or more

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

  • Compare the statistic with the critical value you have determined

    • If $X^2$ > critical value (or $p<\alpha$) then there is sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

    • If $X^2$ < critical value (or $p>\alpha$) then there is insufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

• STEP 6: Write your conclusion

  • If you reject H₀

    • A discrete uniform distribution is not a suitable model

  •  If you do not reject H₀

    • A discrete uniform distribution is a suitable model

  • Be sure to state your conclusion in the context of the question

Chi squared for binomial

How do I do a chi-squared test with a binomial distribution?

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

• For a random variable $X$ to have a binomial distribution:

  • the number of trials ($n$) must be fixed in each observation

  • the trials must be independent

  • each trial can have only two outcomes (success and failure)

  • the probability of success ($p$) must be constant

•  A question may give a precise binomial distribution $\mathrm{B}\left(n,p\right)$ to test 

  • with an assumed value for $p$

•  Or you may be asked to test whether a binomial distribution is suitable without being given an assumed value for $p$ 

  • In this case you will have to calculate an estimate for the value of $p$ for the binomial distribution

  • For $N$ observations of the variable

$p=\frac{\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{successes}}{\mathrm{number}\mathrm{of}\mathrm{trials}\times N}=\frac{\sum \left(x\times f\right)}{n\times N}$

• $f$ is the frequency for each value of $x$ (these are given in a table in the question)

• $n$ is from $\mathrm{B}\left(n,p\right)$ and $N$ is the sum of the observed values

  • Remember that estimating this parameter uses up one degree of freedom

What are the steps?

• STEP 1: Write the hypotheses

  • ${\mathrm{H}}_0$ : A binomial distribution is a suitable model for Variable X

  • ${\mathrm{H}}_1$ : A binomial distribution is not a suitable model for Variable X

    • The hypotheses should always be stated in the context of the question

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

    • If you are given the assumed value of $p$ then state the precise distribution $\mathrm{B}(n,p)$

• STEP 2: Calculate the expected frequencies

  • If you were not given the assumed value of p then you will first have to estimate it using the observed data

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

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

  • You will have to combine rows/columns if any expected values are less than 5 until they are greater than 5

• STEP 3: Calculate the degrees of freedom for the test

  • For $k$ outcomes (after combining expected values if needed)

  • Degrees of freedom is 

    • $\nu =k-1$ if you were given the assumed value of $p$ in the question

    • $\nu =k-2$ if you had to estimate the value of $p$ using data in the question

• STEP 4: Calculate $X^2$ using either version of the formula

$X^2=\sum _{i=1}^n\frac{{\left(O_i-E_i\right)}^2}{E_i}=\left(\sum _{i=1}^n\frac{{O_i}^2}{E_i}\right)-N$ 

• Determine the appropriate ${\chi }^2$ critical value

  • ${\chi }_{\nu }^2\left(\alpha \%\right)$ is the critical value with $\nu$ degrees of freedom for significance level $\alpha$ 

  • use the 'Percentage Points of the ${\chi }^2$ Distribution' table in the exam formula booklet

• Or, alternatively, use a calculator to find the ${\chi }_{\nu }^2$ p-value

  • This is the probability of obtaining a chi-squared value of $X^2$ or more

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

  • Compare the statistic with the critical value you have determined

    • If $X^2$ > critical value (or $p<\alpha$) then there is sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

    • If $X^2$ < critical value (or $p>\alpha$) then there is insufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

• STEP 6: Write your conclusion

  • If you reject H₀

    • A binomial distribution is not a suitable model

  •  If you do not reject H₀

    • A binomial distribution is a suitable model

  • Be sure to state your conclusion in the context of the question

Chi squared for Poisson

How do I do a chi-squared test with a Poisson distribution?

• A chi-squared (χ²) goodness of fit test can be used to test data from a sample suggesting that the population has a Poisson distribution

• For a random variable $X$ to have a Poisson distribution:

  • events must occur independently of each other

  • events must occur singly and randomly

  • events must occur at a constant rate (in space or time)

  • the mean and the variance must be equal

•  You will either be given a precise Poisson distribution $\mathrm{Po}\left(\lambda \right)$ to test 

  • with an assumed value for $\lambda$

•  Or you will be asked to test whether a Poisson distribution is suitable without being given an assumed value for $\lambda$ 

  • In this case you will have to calculate an estimate for the value of $\lambda$ for the Poisson distribution

  • The estimate for $N$ observations is just the mean of the observed sample:

$\lambda =\frac{\sum \left(x\mathit{\times }f\right)}{N}$ 

• $f$is the frequency for each value of $x$ (these are given in a table in the question)

• $N$ is the sum of the observed values

  • Remember that estimating this parameter uses up one degree of freedom

What are the steps?

• STEP 1: Write the hypotheses

  • ${\mathrm{H}}_0$: A Poisson distribution is a suitable model for Variable X

  • ${\mathrm{H}}_1$: A Poisson distribution is not a suitable model for Variable X

    • The hypotheses should always be stated in the context of the question

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

    • If you are given the assumed value of $\lambda$ then state the precise distribution $\mathrm{Po}\left(\lambda \right)$

• STEP 2: Calculate the expected frequencies

  • If you were not given the assumed value of $\lambda$ then you will first have to estimate it using the observed data

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

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

  • Poisson variables start on $X=0$ and go up to infinity

    • If a is the smallest observed value in the table then calculate all of $\mathrm{P}(X\le a)$ for that column

    • If b is the largest observed value in the table then calculate all of $\mathrm{P}(X\ge b)$ up to infinity

      • $1-\mathrm{P}\left(X\le b-1\right)$

  • You will have to combine rows/columns if any expected values are less than 5 until they are greater than 5

• STEP 3: Calculate the degrees of freedom for the test

  • For k outcomes (after combining expected values if needed)

  • Degrees of freedom is 

    • $\nu =k-1$ if you were given the assumed value of $\lambda$

    • $\nu =k-2$ if you had to estimate the value of $\lambda$

• STEP 4: Calculate $X^2$ using either version of the formula

$X^2=\sum _{i=1}^n\frac{{\left(O_i-E_i\right)}^2}{E_i}=\left(\sum _{i=1}^n\frac{{O_i}^2}{E_i}\right)-N$ 

• Determine the appropriate ${\chi }^2$ critical value

  • ${\chi }_{\nu }^2\left(\alpha \%\right)$ is the critical value with $\nu$ degrees of freedom for significance level $\alpha$ 

  • use the 'Percentage Points of the ${\chi }^2$ Distribution' table in the exam formula booklet

  • Or, alternatively, use a calculator to find the ${\chi }_{\nu }^2$ p-value

    • This is the probability of obtaining a chi-squared value of $X^2$ or more

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

  • Compare the statistic with the critical value you have determined

    • If $X^2$ > critical value (or $p<\alpha$) then there is sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

    • If $X^2$ < critical value (or $p>\alpha$) then there is insufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

• STEP 6: Write your conclusion

  • If you reject H₀

    • A Poisson distribution is not a suitable model

  •  If you do not reject H₀

    • A Poisson distribution is a suitable model

  • Be sure to state your conclusion in the context of the question

Chi squared for geometric

How do I do a chi-squared test with a geometric distribution?

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

• For a random variable $X$ to have a geometric distribution:

  • the trials must be independent

  • each trial can have only two outcomes (success and failure)

  • trials are repeated until the first success

  • the probability of success ($p$) must be constant

  • the value of the variable is the number of trials until the first success

•  A question may give a precise geometric distribution $\mathrm{Geo}\left(p\right)$ to test 

  • with an assumed value for $p$

•  Or you may be asked to test whether a geometric distribution is suitable without being given an assumed value for $p$ 

  • In this case you will have to calculate an estimate for the value of $p$ for the geometric distribution

  • For $N$ observations of the variable

$p=\frac{\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{successes}}{\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{N}{\sum \left(x\times f\right)}$

• $f$is the frequency for each value of $x$ (these are given in a table in the question)

• $N$ is the sum of the observed values

  • Remember that estimating this parameter uses up one degree of freedom

What are the steps?

• STEP 1: Write the hypotheses

  • ${\mathrm{H}}_0$ : A geometric distribution is a suitable model for Variable X

  • ${\mathrm{H}}_1$ : A geometric distribution is not a suitable model for Variable X

    • The hypotheses should always be stated in the context of the question

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

    • If you are given the assumed value of $p$ then state the precise distribution $\mathrm{Geo}\left(p\right)$

• STEP 2: Calculate the expected frequencies

  • If you were not given the assumed value of p then you will first have to estimate it using the observed data

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

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

  • Geometric variables start on 1 and go up to infinity

    • If a is the smallest observed value in the table then calculate all of $X\le a$  for that column

    • If b is the largest observed value in the table then calculate all of $X\ge b$ up to infinity

      • The formulae $\mathrm{P}\left(X\le x\right)=1-{\left(1-p\right)}^x$ and $\mathrm{P}\left(X\ge x\right)={\left(1-p\right)}^{x-1}$ can help

  • You will have to combine rows/columns if any expected values are less than 5 until they are greater than 5

• STEP 3: Calculate the degrees of freedom for the test

  • For $k$ outcomes (after combining expected values if needed)

  • Degrees of freedom is 

    • $\nu =k-1$ if you were given the assumed value of $p$

    • $\nu =k-2$ if you had to estimate the value of $p$

• STEP 4: Calculate $X^2$ using either version of the formula

$X^2=\sum _{i=1}^n\frac{{\left(O_i-E_i\right)}^2}{E_i}=\left(\sum _{i=1}^n\frac{{O_i}^2}{E_i}\right)-N$ 

• Determine the appropriate ${\chi }^2$ critical value

  • ${\chi }_{\nu }^2\left(\alpha \%\right)$ is the critical value with $\nu$ degrees of freedom for significance level $\alpha$ 

  • use the 'Percentage Points of the ${\chi }^2$ Distribution' table in the exam formula booklet

• Or, alternatively, use a calculator to find the ${\chi }_{\nu }^2$ p-value

  • This is the probability of obtaining a chi-squared value of $X^2$ or more

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

  • Compare the statistic with the critical value you have determined

    • If $X^2$ > critical value (or $p<\alpha$) then there is sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

    • If $X^2$ < critical value (or $p>\alpha$) then there is insufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$

• STEP 6: Write your conclusion

  • If you reject H₀

    • A geometric distribution is not a suitable model

  •  If you do not reject H₀

    • A geometric distribution is a suitable model

  • Be sure to state your conclusion in the context of the question