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

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Naomi C

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Maths

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 ( $n = 60$ ) 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 $\sum_{}^{} observed = \sum_{}^{} expected = n = 60$

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

The null hypothesis, $H_{0}$ , is the assumption that there is no difference between the expected distribution and the actual distribution
- e.g. $H_{0} :$ The dice outcomes have a uniform distribution with the true probability of each outcome being $\frac{1}{6}$
  - It is assumed to be correct, unless evidence proves otherwise
The alternative hypothesis, $H_{a}$ , is the assumption that there is a difference between the expected distribution and the actual distribution
- e.g. $H_{a} :$ The dice outcomes do not have a uniform distribution with the true probability of each outcome being $\frac{1}{6}$

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, $X^{2}$ , can be calculated from the formula given to you in the exam
- $χ^{2} = \sum_{}^{} \frac{{(observed - expected)}^{2}}{expected}$
The larger $X^{2}$ 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 $6 - 1 = 5$

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, $α %$ 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, $X^{2}$ , to the critical value from the table
- If $X^{2} > critical value$ then the null hypothesis should be rejected
  - The expected distribution is not a suitable model for the data
- If $X^{2} < critical value$ 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 $X^{2}$ , 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, $α$ , with the calculator's $p$ -value
  - If using the $p$ -value, remember
    - $p < α$ , reject the null hypothesis
    - $p > α$ , do not reject the null hypothesis

Exam Tip

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 $X^{2}$ value and the degrees of freedom.

If you compare the $p$ -value with $α$ , don't forget that the inequalities are the opposite to when you are comparing the $X^{2}$ 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

$H_{0} :$ The actual distribution of points awarded is the same as the expected distribution of points awarded

$H_{a} :$ The 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 $α = 0.05$

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, $X^{2}$ , $\sum \frac{{(observed - expected)}^{2}}{expected}$

$\begin{array}{rcl} X^{2} & = & \frac{{(69 - 60)}^{2}}{60} + \frac{{(13 - 20)}^{2}}{20} + \frac{{(11 - 15)}^{2}}{15} + \frac{{(7 - 5)}^{2}}{5} \\ = & \frac{81}{60} + \frac{49}{20} + \frac{16}{15} + \frac{4}{5} \\ = & \frac{17}{3} \\ = & 5.666 . . . \end{array}$

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 $α = 0.05$

$critical value = 12.84$

Compare the calculated $X^{2}$ value to the critical value and state the conclusion of the test

$\begin{array}{rcl} 5.666 . . . & < & 12.84 \\ X^{2} & < & critical value \end{array}$

$H_{0}$ 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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