Goodness-of-Fit Procedure
Calculating the Test Statistic
The Problem
Example #4: Let us have a 6-sided die. Let us, furthermore, hypothesize that the die is fair. To test this hypothesis, we collect data. In this case, that data consists of rolling that die 46 times and recording which face came up each time. Here is the data summarized in tabular form:
| Die Face | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 5 | 6 | 10 | 8 | 7 | 10 |
With this information, calculate the test statistic corresponding to the null hypothesis that this die is fair. Note that the faces of a fair die are equally likely. Thus, \(p_i = 1/g = 1/ 6 = 0.166667\).
Information given:
To summarize the above, the values of import are:
| \( x_1 \) | = | 5 |
|---|---|---|
| \( x_2 \) | = | 6 |
| \( x_3 \) | = | 10 |
| \( x_4 \) | = | 8 |
| \( x_5 \) | = | 7 |
| \( x_6 \) | = | 10 |
| \( \mu_1 \) | = | 7.666667 |
| \( \mu_2 \) | = | 7.666667 |
| \( \mu_3 \) | = | 7.666667 |
| \( \mu_4 \) | = | 7.666667 |
| \( \mu_5 \) | = | 7.666667 |
| \( \mu_6 \) | = | 7.666667 |
It may be helpful if you calculate these values yourself. Once you have, you can check your answers by hovering your mouse over the grey spaces to see if you calculated them correctly.
Your Answer
In the box below, please enter the test statistic for the null hypothesis and the data given above. Once you have done so, click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.
![[Correct!]](images/correct.png "You are correct. Well done!")
You got the correct answer of X2 = 2.7826. Congratulations!
![[Incorrect!]](images/incorrect.png "You are not correct.")
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