Chapter 12: Q. 12.20 (page 491)

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

Expert verified

The variables differ from the given distribution.

Step by step solution

Step 1. Given Information

We are given,

The sample size is,

$n = 29 + 13 + 5 + 25 + 28 = 100$

Level of significance, $α = 0.10$

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	$\frac{(O b s - E x p)^{2}}{E x p}$
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,

$\sum \frac{(O b s - Exp)^{2}}{Exp} = 8.417$

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

$= 5 - 1 = 4$

Critical value is $χ_{0.10}^{2}$ for $4 d f$ is $7.779$

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

$χ^{2} = 8.417 > χ_{0.10}^{2} = 7.779$ ,

because it fall in the rejection region. So, we reject the null hypothesis, $H_{0}$

Reject $H_{0}; H_{0}$ Variable has distribution given in the problem.

Therefore, the variables differ from the given distribution.

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Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation the goodness of fit .

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

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