4.7.3 Goodness of Fit Test

Chi-Squared GOF: Uniform

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

A chi-squared ( $χ^{2}$ ) 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 $ν = k - 1$
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 $χ_{c a l c}^{2}$
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

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 $χ^{2}$ 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.

Find the expected frequency of sales for each week if the data were uniformly distributed.

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Write down the null and alternative hypotheses.

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Write down the number of degrees of freedom for this test.

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Calculate the p-value.

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State the conclusion of the test. Give a reason for your answer.

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Chi-Squared GOF: Binomial

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

A chi-squared ( $χ^{2}$ ) 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 $B (n, p)$
- H₁: Variable X cannot be modelled by the binomial distribution $B (n, p)$
  - 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 the outcome using the binomial distribution $P (X = x)$
- Multiply the probability by the total frequency $P (X = x) \times N$
STEP 3: Calculate the degrees of freedom for the test
- For k outcomes
- Degrees of freedom is $ν = k - 1$
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 $χ_{c a l c}^{2}$
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 $B (n, p)$
  - Therefore this suggests that the data does not follow $B (n, p)$
- If you accept H₀
  - There is insufficient evidence to suggest that variable X does not follow the binomial distribution $B (n, p)$
  - Therefore this suggests that the data follows $B (n, p)$

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 $χ^{2}$ 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.

State the null and alternative hypotheses.

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Assuming the binomial distribution holds, find the expected number of people that would defeat exactly one boss.

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Calculate the p-value for the test.

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State the conclusion of the test. Give a reason for your answer. opxxE5_K_4-7-3-ib-ai-sl-gof-binomial-d-we-solution

Chi-Squared GOF: Normal

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

A chi-squared ( $χ^{2}$ ) 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 $N (μ, σ^{2})$
- H₁: Variable X cannot be modelled by the normal distribution $N (μ, σ^{2})$
  - 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 the outcome using the normal distribution $P (a < X < b)$
  - Beware of unbounded inequalities $P (X < b)$ or $P (X > a)$ for the class intervals on the 'ends'
- Multiply the probability by the total frequency $P (a < X < b) \times N$
STEP 3: Calculate the degrees of freedom for the test
- For k class intervals
- Degrees of freedom is $ν = k - 1$
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 $χ_{c a l c}^{2}$
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 $N (μ, σ^{2})$
  - Therefore this suggests that the data does not follow $N (μ, σ^{2})$
- If you accept H₀
  - There is insufficient evidence to suggest that variable X does not follow the normal distribution $N (μ, σ^{2})$
  - Therefore this suggests that the data follows $N (μ, σ^{2})$

Worked example

300 marbled ducks in Quacktown are weighed and the results are shown in the table below.

Mass (g)	Frequency
$m < 470$	10
$470 \leq m < 520$	158
$520 \leq m < 570$	123
$m \geq 570$	9

A $χ^{2}$ 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.

Calculate the expected frequencies, giving your answers correct to 2 decimal places.

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