Sam was playing a dice game. The rules of the game were: for each turn, the player rolls three dice and wins only if three sixes are rolled. After ten failures, Sam suspected that the dice may have been tampered with. Sam stole the dice and brought them home, where he rolled them 1000 times  the observed sample.
He also had three dice he knew to be fair, and rolled those 1000 times  this is the expected sample. For each roll he recorded the number of sixes. The results were as follows:

observed

expected

0 sixes

580

570

1 six

354

360

2 sixes

63

64

3 sixes

3

6



Now he wants to test if the dice are fair or not.
1.

Determine the null hypothesis:


Null Hypothesis: The three dice are fair. (Observed sample dose not differ from expected sample.)

2.

Substitute the information into the formula:


$x=\frac{{\left(580570\right)}^{2}}{570}+\frac{{\left(354360\right)}^{2}}{360}+\frac{{\left(6364\right)}^{2}}{64}+\frac{{\left(36\right)}^{2}}{6}=1.791063596$


$p\mathrm{value}=\mathrm{Probability}\left(X\u02c62>1.791063596\right)=0.616881663760937$, $X\u02c62\u02dc\mathrm{ChiSquare}\left(3\right)$


This statistical test does not provide enough evidence to conclude that the null hypothesis is false, so we fail to reject the null hypothesis.
