Goodness of Fit Test (DP IB Maths: AI SL)
Revision Note
Author
DanExpertise
Maths
Chi-Squared GOF: Uniform
What is a chi-squared goodness of fit test for a given distribution?
- A chi-squared () 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
- 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
- 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
- EITHER compare the χ² statistic with the given critical value
- 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
- If you reject H0
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.
a)
Find the expected frequency of sales for each week if the data were uniformly distributed.
b)
Write down the null and alternative hypotheses.
c)
Write down the number of degrees of freedom for this test.
d)
Calculate the p-value.
e)
State the conclusion of the test. Give a reason for your answer.
Chi-Squared GOF: Binomial
What is a chi-squared goodness of fit test for a binomial distribution?
- A chi-squared () 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
- H1 : 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 expected frequencies
- Find the probability of the outcome using the binomial distribution
- Multiply the probability by the total frequency
- STEP 3: Calculate the degrees of freedom for the test
- For k outcomes
- Degrees of freedom is
- 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
- 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
- EITHER compare the χ² statistic with the given critical value
- STEP 6: Write your conclusion
- If you reject H0
- 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 H0
- 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 H0
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.
a)
State the null and alternative hypotheses.
b)
Assuming the binomial distribution holds, find the expected number of people that would defeat exactly one boss.
c)
Calculate the p-value for the test.
d)
State the conclusion of the test. Give a reason for your answer.
Chi-Squared GOF: Normal
What is a chi-squared goodness of fit test for a normal distribution?
- A chi-squared () 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
- H1 : 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 expected frequencies
- Find the probability of the outcome using the normal distribution
- Beware of unbounded inequalities or for the class intervals on the 'ends'
- Multiply the probability by the total frequency
- Find the probability of the outcome using the normal distribution
- STEP 3: Calculate the degrees of freedom for the test
- For k class intervals
- Degrees of freedom is
- 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
- 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
- EITHER compare the χ² statistic with the given critical value
- STEP 6: Write your conclusion
- If you reject H0
- 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 H0
- 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 H0
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.
a)
Calculate the expected frequencies, giving your answers correct to 2 decimal places.
b)
Write down the null and alternative hypotheses.
c)
Calculate the statistic.
d)
Given that the critical value is 6.251, state the conclusion of the test. Give a reason for your answer.
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