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:$0.5,0.3,0.2$

Observed frequencies:$147,115,88$

Significance level$=0.01$

Short answer

The variable has the given specified distribution.

Step-by-step solution

Step 1. Given Information 

We are given the sample size is,

$$\begin{gathered} n=9+7+1+12+21 \\ =50 \end{gathered}$$

Level of significance,

$\alpha =0.01$

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	$\frac{(Obs-Exp{)}^2}{Exp}$
9	0.2	10	-1	0.1
7	0.1	5	2	0.8
1	0.1	5	-4	3.2
12	0.3	15	-3	0.6
21	0.3	15	-6	2.4

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

We get,

$\sum \frac{(Obs-Exp{)}^2}{Exp}=7.1$

The degrees of freedom of the given data is,

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

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

The value of the test statistic is,${\chi }^2=7.1$

${\chi }^2=7.1<{\chi }_{0.10}^2=7.779$,

because it does not fall in the rejection region.

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

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

Therefore, the variable has the given specified distribution.