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AP®StatisticsCollege BoardStudy GuidesUnits 6-9: InferenceGoodness of Fit (Chi-Square)Hypothesis Tests for Goodness of Fit

Hypothesis Tests for Goodness of Fit (College Board AP® Statistics)

Study Guide

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Goodness of fit test

What is a goodness of fit test?

  • A goodness of fit test is a measure of how well real-life observed data fits a theoretical model

    • For example, from a coin being thrown a large number of times, a random sample of 20 throws may be inspected 

      • You may observe 13 heads

      • You would expect 10 heads

  • Observed and expected values can be shown in a table

    • For example, a fair dice being rolled a large number of times and a randomly selected sample of 60 rolls (n equals 60) could be displayed as below

Outcome

1

2

3

4

5

6

Observed

12

7

8

10

14

9

Expected

10

10

10

10

10

10

  • Note that sum from blank to blank of observed equals sum from blank to blank of expected equals n equals 60

What are the null and alternative hypotheses for a goodness of fit test?

  • The null hypothesis, straight H subscript 0, is the assumption that there is no difference between the expected distribution and the actual distribution

    • e.g. straight H subscript 0 space colon The dice outcomes have a uniform distribution with the true probability of each outcome being 1 over 6

      • It is assumed to be correct, unless evidence proves otherwise

  • The alternative hypothesis, straight H subscript straight a, is the assumption that there is a difference between the expected distribution and the actual distribution

    • e.g. straight H subscript straight a colon The dice outcomes do not have a uniform distribution with the true probability of each outcome being 1 over 6

What are the conditions for a goodness of fit test?

  • When performing a chi-square goodness of fit test:

    • Observed values must come from a random sample

    • Observed values must be independent

      • They are sampled with replacement

      • or the sample size is less than 10% of the population size

    • Expected values must meet the large counts condition

      • Each expected value must be greater than or equal to 5

How do I calculate a chi-square value?

  • The chi-square value for the goodness of fit test, X squared, can be calculated from the formula given to you in the exam

    • chi squared equals sum from blank to blank of open parentheses observed minus expected close parentheses squared over expected

  • The larger X squared is, the more different the observed values are from the expected values

What are degrees of freedom?

  • The number of degrees of freedom, 'dof', is equal to

    • the number of possible outcomes subtract 1

    • e.g. dof for the fair die is 6 minus 1 equals 5

How do I use the chi-square distribution table?

  • You can use the chi-square tables given to you in the exam to find the critical value

    • This is the threshold value that determines whether you reject the null hypothesis or not

  • To find critical value from the tables, you need the significance level, alpha percent sign and the dof

    • The critical value is located in the cell where the relevant row and column intersect

How do I conclude a hypothesis test?

  • Conclusions to a hypothesis test need to show two things:

    • a decision about the null hypothesis

    • an interpretation of this decision in the context of the question

  • To make the decision, compare the calculated goodness of fit value, X squared, to the critical value from the table

    • If X squared greater than critical space value then the null hypothesis should be rejected

      • The expected distribution is not a suitable model for the data

    • If X squared less than critical space value then the null hypothesis should not be rejected

      • This means there is no difference between the observed and expected distributions

      • The expected distribution is a suitable model for the data

How can I perform a chi-square test on the calculator?

  • To complete a chi-square test on your calculator:

    • Add the expected values and the observed values into a table

    • Perform a chi-square goodness of fit test

    • Compare your calculated X squared, with the critical value from the chi-square tables

  • Another calculator method is to:

    • Add the expected values and the observed values into a table

    • Perform a chi-square goodness of fit test

    • Compare the given significance level, alpha, with the calculator's p-value

      • If using the p-value, remember

        • p less than alpha, reject the null hypothesis

        • p greater than alpha, do not reject the null hypothesis

Examiner Tips and Tricks

Even if you perform the chi-square goodness of fit test on your calculator, it is still important to show all of your working to demonstrate full understanding. Therefore you should still calculate the X squared value and the degrees of freedom.

If you compare the p-value with alpha, don't forget that the inequalities are the opposite to when you are comparing the X squared value to the critical value when you are determining whether or not to reject the null hypothesis!

Worked Example

A game is meant to award points according to the probability distribution below.

Points

2

4

8

10

Probability

0.6

0.2

0.15

0.05

The game is played by 100 people who are randomly selected, giving the results below.

Points

2

4

8

10

Frequency

69

13

11

7

Test, at the 5% level of significance, whether or not the game is operating correctly.

Write the null and alternative hypotheses

straight H subscript 0 space colon The actual distribution of points awarded is the same as the expected distribution of points awarded

straight H subscript straight a space colon spaceThe actual distribution of points awarded is different to the expected distribution of points awarded

State the type of test being used

The correct inference procedure is a chi-square goodness of fit test at alpha equals 0.05

Calculate the expected values by multiplying each probability by the number of players (100)

Points

2

4

8

10

Probability

0.6

0.2

0.15

0.05

Expected

60

20

15

5

Verify the conditions for the test

All conditions for inference have been met:

  • The sample of players is randomly selected

  • All expected values are greater than or equal to 5

Calculate the chi-square value, X squared, sum open parentheses observed minus expected close parentheses squared over expected

table row cell X squared end cell equals cell open parentheses 69 minus 60 close parentheses squared over 60 plus open parentheses 13 minus 20 close parentheses squared over 20 plus open parentheses 11 minus 15 close parentheses squared over 15 plus open parentheses 7 minus 5 close parentheses squared over 5 end cell row blank equals cell 81 over 60 plus 49 over 20 plus 16 over 15 plus 4 over 5 end cell row blank equals cell 17 over 3 end cell row blank equals cell 5.666... end cell end table

State the number of degrees of freedom

degrees of freedom = 4 - 1 = 3

Find the critical value from the chi-square tables

Find the row corresponding to 3 degrees of freedom and the column corresponding to alpha equals 0.05

critical space value equals 12.84

Compare the calculated X squared value to the critical value and state the conclusion of the test

table row cell 5.666... end cell less than cell 12.84 end cell row cell X squared end cell less than cell critical space value end cell end table

straight H subscript 0 is not rejected

Interpret the result in the context of the question

We do not have sufficient evidence to suggest that the observed proportions of the points awarded do not come from the same distribution as is given by the expected proportions in the table above

The game appears to be working correctly

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    Previous:The Chi-Square DistributionNext:Tests for Independence
    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.