Question

Loaded Die? A gambler thinks a die may be loaded, that is, that the six numbers are not equally likely. To test his suspicion, he rolled the die 150 times and obtained the data shown in the following table.

Number	1	2	3	4	5	6
Frequency	23	26	23	21	31	26

Do the data provide sufficient evidence to conclude that the die is loaded? Perform the hypothesis test at the 0.05 level of significance.

Short answer

Ans: The data do not provide sufficient evidence to conclude that the die is loaded at $0.05$of significance.

Step-by-step solution

Step 1. Given information.

given,

Number	1	2	3	4	5	6
Frequency	23	26	23	21	31	26

Step 2. First, we will test whether the data provide sufficient evidence to indicate that the die is loaded.

The test hypothesis:

$H_0$: The data provides sufficient evidence to show that the dice were not loaded, that is, all six numbers have equal probability.

$H_a$: The data provide sufficient evidence to indicate that the die is loaded

Calculating the Goodness of fit:

Number	Observed frequencies	Relative frequencies	Expected frequencies	Obs- Exp	$\frac{{(Obs-Exp)}^2}{Exp}$
1	23	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$	25	-2	0.16
2	26	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$	25	1	0.04
3	23	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$	25	-2	0.16
4	21		25	-4	0.64
5	31	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$	25	6	1.44
6	26	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$	25	1	0.04

$\sum \frac{(Obs-Exp{)}^2}{\mathrm{Exp}}=2.48$

Step 3. The degrees of freedom of the given data are, 

$$\begin{gathered} =k-1 \\ =6-1 \\ =5 \end{gathered}$$

Critical value is ${\chi }_{0.05}^{2}5df\text{is}11.070$

The value of the test statistic is,

$X^2=2.48$

${\chi }^2=2.48<{\chi }_{0.05}^2=11.070$, because it does not fall in the rejection region.

Thus, we do not reject the null hypothesis, $H_0$

Therefore, we do not reject localid="1652033723048" $H_0:H_0$Variable distribution of the given problem.

Therefore, the data provide sufficient evidence to show that the dice were not loaded, that is, all six numbers have the same probability.

And, The data do not provide sufficient evidence to show that deaths were loaded at a 5% value.

localid="1652033609570" $\begin{array}{r}=11.070\\ {\chi }^2<11.070\\ p=0.780>0.05\\ \mathrm{So},{\mathrm{H}}_{\mathrm{O}}\text{should be rejected.}\end{array}$

The data do not provide sufficient evidence to conclude that the die is loaded at $0.05$of significance.