ChiSquared GOF: Uniform
What is a chisquared goodness of fit test for a uniform distribution?
 A chisquared () goodness of fit test is used to test data from a sample which suggests that the population has a uniform distribution
 This means all outcomes are equally likely
What are the steps for a chisquared goodness of fit test for a uniform distribution?
 STEP 1: Write the hypotheses
 H_{0 }: Variable X can be modelled by a uniform distribution
 H_{1 }: Variable X cannot be modelled by a uniform distribution
 Make sure you clearly write what the variable is and don’t just call it X
 STEP 2: Calculate the degree of freedom for the test
 For k outcomes
 Degree of freedom is
 STEP 3: Calculate the expected frequencies
 Divide the total frequency N by the number of outcomes k
 STEP 4: Enter the frequencies and the degree 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 pvalue
 The χ² statistic is denoted as
 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 H_{0}
 If χ² statistic < critical value then accept H_{0}
 OR compare the pvalue with the given significance level
 If pvalue < significance level then reject H_{0}
 If pvalue > significance level then accept H_{0}
 EITHER compare the χ² statistic with the given critical value
 STEP 6: Write your conclusion
 If you reject H_{0}
 There is sufficient evidence to suggest that variable X does not follow a uniform distribution
 Therefore this suggests that the data is not uniformly distributed
 If you accept H_{0}
 There is insufficient evidence to suggest that variable X does not follow a uniform distribution
 Therefore this suggests that the data is uniformly distributed
 If you reject H_{0}
Worked Example
A car salesman is interested in how his sales are distributed and records his sales results over a period of six weeks. The data is shown in the table.
Week 
1 
2 
3 
4 
5 
6 
Number of sales 
15 
17 
11 
21 
14 
12 
A goodness of fit test is to be performed on the data at the 5% significance level to find out whether the data fits a uniform distribution.
ChiSquared GOF: Binomial
What is a chisquared goodness of fit test for a binomial distribution?
 A chisquared () 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 chisquared goodness of fit test for a binomial distribution?
 STEP 1: Write the hypotheses
 H_{0 }: Variable X can be modelled by the binomial distribution
 H_{1 }: Variable X cannot be modelled by the binomial distribution
 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 degrees of freedom for the test
 For k outcomes
 Degree of freedom is
 STEP 3: Calculate the expected frequencies
 Find the probability of the outcome using the binomial distribution
 Multiply the probability by the total frequency
 STEP 4: Enter the frequencies and the degree 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 pvalue
 The χ² statistic is denoted as
 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 H_{0}
 If χ² statistic < critical value then accept H_{0}
 OR compare the pvalue with the given significance level
 If pvalue < significance level then reject H_{0}
 If pvalue > significance level then accept H_{0}
 EITHER compare the χ² statistic with the given critical value
 STEP 6: Write your conclusion
 If you reject H_{0}
 There is sufficient evidence to suggest that variable X does not follow the binomial distribution
 Therefore this suggests that the data does not follow
 If you accept H_{0}
 There is insufficient evidence to suggest that variable X does not follow the binomial distribution
 Therefore this suggests that the data follows
 If you reject H_{0}
Worked Example
A stage in a video game has three boss battles. 1000 people try this stage of the video game and the number of bosses defeated by each player is recorded.
Number of bosses defeated 
0 
1 
2 
3 
Frequency 
490 
384 
111 
15 
A goodness of fit test at the 5% significance level is used to decide whether the number of bosses defeated can be modelled by a binomial distribution with a 20% probability of success.
ChiSquared GOF: Normal
What is a chisquared goodness of fit test for a normal distribution?
 A chisquared () 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 chisquared goodness of fit test for a normal distribution?
· STEP 1: Write the hypotheses

 H_{0 }: Variable X can be modelled by the normal distribution
 H_{1 }: Variable X cannot be modelled by the normal distribution
 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 degrees of freedom for the test
 For k outcomes
 Degree of freedom is
 STEP 3: Calculate the expected frequencies
 Find the probability of the outcome using the normal distribution
 Beware of unbounded inequalities or
 Multiply the probability by the total frequency
 Find the probability of the outcome using the normal distribution
 STEP 4: Enter the frequencies and the degree 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 pvalue
 The χ² statistic is denoted as
 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 H_{0}
 If χ² statistic < critical value then accept H_{0}
 OR compare the pvalue with the given significance level
 If pvalue < significance level then reject H_{0}
 If pvalue > significance level then accept H_{0}
 EITHER compare the χ² statistic with the given critical value
 STEP 6: Write your conclusion
 If you reject H_{0}
 There is sufficient evidence to suggest that variable X does not follow the normal distribution
 Therefore this suggests that the data does not follow
 If you accept H_{0}
 There is insufficient evidence to suggest that variable X does not follow the normal distribution
 Therefore this suggests that the data follows
 If you reject H_{0}
Worked Example
300 marbled ducks in Quacktown are weighed and the results are shown in the table below.
Mass (g) 
Frequency 
10 

158 

123 

9 
A goodness of fit test at the 10% significance level is used to decide whether the mass of a marbled duck can be modelled by a normal distribution with mean 520 g and standard deviation 30 g.