In each of Exercises 12.18-12.23, we have provided a distribution and the observed frequencies of the values of a variable from a simple random sample of a population. In each case, use the chi-square goodness-of-fit test to decide, at the specified significance level, whether the distribution of the variable differs from the given distribution.

Distribution: 0.2, 0.4, 0.3, 0.1

Observed frequencies: 85, 215, 130, 70

Significance level = 0.05

The sample size is n= 85+215+130+70=500.

Level of significance , $\alpha =0.05$

Now, we want to perform the hypothesis test

The test hypotheses,

${\mathrm{H}}_0$: The variable has the given specified distribution

${\mathrm{H}}_1$: The variable differ from the given distribution

Calculating the Goodness of fit :

Observed frequencies	Relative frequencies	Expected frequencies	Obs - Exp
85	0.2	100	-15	2.25
215	0.4	200	15	1.125
130	0.3	150	-20	2.67
70	0.1	50	20	8
				localid="1651872624059" $\sum _{}\frac{{(\mathrm{Obs}-\mathrm{Exp})}^2}{\mathrm{Exp}}=14.0417$

localid="1651872628579" $\sum _{}\frac{{(\mathrm{Obs}-\mathrm{Exp})}^2}{\mathrm{Exp}}=14.0417$

The degrees of freedom of the given data is, k-1

=4-1 = 3

Critical value is localid="1651872634369" ${{\mathcal{X}}^2}_{0.05}$for 3df is 7.815

The value of the test statistic is, localid="1651872639841" ${\mathcal{X}}^2$= 14.0417

localid="1651872644534" ${\mathcal{X}}^2$=14.0417 > localid="1651872649178" ${{\mathcal{X}}^2}_{0.05}$= 7.815, because it fall in the rejection region So, wo reject the null hypothesis,${\mathrm{H}}_0$

localid="1651872654495" $\therefore \boxed{R\mathrm{eject}{\mathrm{H}}_0},{\mathrm{H}}_0$Variable has distribution given in the problem .

Therefore, the variable differ from given specified distribution.