Question

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: 39, 78, 64, 19

Significance level = 0.05

Short answer

The test hypotheses,

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

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

wo do not reject the null hypothesis,${\mathrm{H}}_0$

$\therefore \boxed{\mathrm{Do}\mathrm{not}\mathrm{reject}{\mathrm{H}}_0},{\mathrm{H}}_0$Variable has distribution given in the problem .

Therefore, the variable has the given specified distribution.

Step-by-step solution

Step 1. Given 

The sample size is n= 39+78+64+19=200.

Level of significance ,$\alpha =0.05$

Step 2. Calculation the goodness of fit .

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	$\frac{{(Obs-Exp)}^2}{Exp}$
39	0.2	40	-1	0.025
78	0.4	80	-2	0.050
64	0.3	60	4	0.267
19	0.1	20	-1	0.050
				$\sum _{}\frac{{(Obs-Exp)}^2}{Exp}=0.392$

$\sum _{}\frac{{(Obs-Exp)}^2}{Exp}=0.392$

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

=4-1 = 3

Critical value is ${{\mathcal{X}}^2}_{0.05}$for 3df is 9.815

The value of the test statistic is, ${\mathcal{X}}^2$=0.392

${\mathcal{X}}^2$=0.392<${{\mathcal{X}}^2}_{0.05}$= 7.815, because it does not fall in the rejection region So, wo do not reject the null hypothesis,${\mathrm{H}}_0$

$\therefore \boxed{\mathrm{Do}\mathrm{not}\mathrm{reject}{\mathrm{H}}_0},{\mathrm{H}}_0$Variable has distribution given in the problem .

Therefore, the variable has the given specified distribution.