Goodness of Fit Test (DP IB Maths: AI SL)

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Dan

Last updated

31 July 2023

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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.

4-7-3-ib-ai-sl-gof-uniform-a-we-solution

Write down the null and alternative hypotheses.

4-7-3-ib-ai-sl-gof-uniform-b-we-solution

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

4-7-3-ib-ai-sl-gof-uniform-c-we-solution

Calculate the p-value.

4-7-3-ib-ai-sl-gof-uniform-d-we-solution

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

4-7-3-ib-ai-sl-gof-uniform-e-we-solution

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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.

4-7-3-ib-ai-sl-gof-binomial-a-we-solution

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

t9ph9q9z_4-7-3-ib-ai-sl-gof-binomial-b-we-solution

Calculate the p-value for the test.

3sGACCT3_4-7-3-ib-ai-sl-gof-binomial-c-we-solution

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

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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.

4-7-3-ib-ai-sl-gof-normal-a-we-solution

Write down the null and alternative hypotheses.

4-7-3-ib-ai-sl-gof-normal-b-we-solution

Calculate the

χ^{2}

statistic.

4-7-3-ib-ai-sl-gof-normal-c-we-solution

Given that the critical value is 6.251, state the conclusion of the test. Give a reason for your answer.

4-7-3-ib-ai-sl-gof-normal-d-we-solution

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Goodness of Fit Test (DP IB Maths: AI SL)

Revision Note

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

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

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

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

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

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

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Exponents & Logarithms

1.1.3 Approximation & Estimation

1.1.4 GDC: Solving Equations

1.2 Sequences & Series

1.2.1 Language of Sequences & Series

1.2.2 Arithmetic Sequences & Series

1.2.3 Geometric Sequences & Series

1.2.4 Applications of Sequences & Series

1.3 Financial Applications

1.3.1 Compound Interest & Depreciation

1.3.2 Amortisation & Annuities

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear & Piecewise Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Voronoi Diagrams

3.4.1 Voronoi Diagrams

3.4.2 Toxic Waste Dump Problem

4. Statistics & Probability

4.1 Statistics Toolkit

4.1.1 Sampling & Data Collection

4.1.2 Statistical Measures

4.1.3 Frequency Tables

4.1.4 Linear Transformations of Data

4.1.5 Outliers

4.1.6 Univariate Data

4.1.7 Interpreting Data

4.2 Correlation & Regression

4.2.1 Bivariate data

4.2.2 Correlation Coefficients

4.2.3 Linear Regression

4.3 Probability

4.3.1 Probability & Types of Events

4.3.2 Conditional Probability

4.3.3 Sample Space Diagrams

4.4 Probability Distributions

4.4.1 Discrete Probability Distributions

4.4.2 Expected Values

4.5 Binomial Distribution

4.5.1 The Binomial Distribution

4.5.2 Calculating Binomial Probabilities

4.6 Normal Distribution

4.6.1 The Normal Distribution

4.6.2 Calculations with Normal Distribution

4.7 Hypothesis Testing