91Ó°ÊÓ

A six-sided die is rolled $120$ times.

The six-sided dice are assumed to be symmetrical. As a result, we anticipate receiving each number from 1 to 6 an equal number of times.

We roll the dice $120$times, so we expect to get each number $\frac{120}{6}=20$times.

The table with data is given by

$\begin{array}{|ccc|}\hline \text{Face Value}& \text{Frequency}& \text{Expected}\\ 1& 15& 20\\ 2& 29& 20\\ 3& 16& 20\\ 4& 15& 20\\ 5& 30& 20\\ 6& 15& 20\\ \hline\end{array}$

We want to test these hypothesis:

$H_0$: The die is fair

$H_1$: The die is not fair

There are six numbers of one dice, thus the number of degrees of freedom is $6-1=5$.

We are using the ${\mathrm{χ}}^2$distribution. Test statistic is given by:

${\mathrm{χ}}^2=\frac{(15-20{)}^2}{20}+\frac{(29-20{)}^2}{20}+\frac{(16-20{)}^2}{20}+\frac{(15-20{)}^2}{20}+\frac{(30-20{)}^2}{20}+\frac{(15-20{)}^2}{20}$

$=\frac{5^2+9^2+4^2+5^2+{10}^2+5^2}{20}$

$=\frac{272}{20}$

$=13.6$

Using the applet, we can see that p-value is $0.01836$:

Taking $\mathrm{Î\pm }=0.05$, we can see that

$0.05>0.01836=p$

This signifies that the null hypothesis is ruled out. There is evidence to suggest that the dice aren't balanced.