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: $0.2,0.1,0.1,0.3,0.3$

Observed frequencies: $29,13,5,25,28$

Significance level $=0.10$

Short answer

The variables differ from the given distribution.

Step-by-step solution

Step 1. Given Information

We are given,

The sample size is,

$$\begin{gathered} n=29+13+5+25+28 \\ =100 \end{gathered}$$

Level of significance,$\alpha =0.10$

Step 2. Calculation the goodness of fit . 

Performing the hypothesis test,

The test hypotheses,

$H_0$:The variable has the given specified distribution

$H_a$:The variable differ from the given distribution

Calculating the Goodness of fit,

Observed frequencies	Relative frequencies	Expected frequencies	Obs-Exp	$\frac{(Obs-Exp{)}^2}{Exp}$
29	0.2	20	9	4.05
13	0.1	10	3	0.90
5	0.1	10	-5	2.50
25	0.3	30	-5	0.83
28	0.3	3	-2	0.13

Step 3. Finding the distribution of variables differs or not. 

We get,

$\sum \frac{(Obs-Exp{)}^2}{Exp}=8.417$

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

$$\begin{gathered} =5-1 \\ =4 \end{gathered}$$

Critical value is ${\chi }_{0.10}^2$for $4df$is$7.779$

The value of the test statistic is,${\chi }^2=8.417$

${\chi }^2=8.417>{\chi }_{0.10}^2=7.779$,

because it fall in the rejection region. So, we reject the null hypothesis,$H_0$

Reject ${\mathrm{H}}_0;{\mathrm{H}}_0$Variable has distribution given in the problem.

Therefore, the variables differ from the given distribution.