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 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: \(45, 39, 16\)

Significance level: \(=0.01\)

Short answer

The variable has given distribution.

Step-by-step solution

Step 1. Given information

Distribution: \(0.5, 0.3, 0.2\)

Observed frequencies: \(45, 39, 16\)

Significance level: \(=0.01\)

Step 2. Calculation

Consider goodness of fit test

The sample size

\(n=45+39+16\)

\(=100\)

Observed frequencies	\(45\)	\(39\)	\(16\)
Relative frequencies	\(0.5\)	\(0.3\)	\(0.2\)
Expected frequencies	\(50\)	\(30\)	\(20\)
\(\frac{(obs-exp)^{2}}{exp}	\(0.5\)	\(2.9\)	\(0.8\)	\(\chi^{2}=\sum \frac{(obs-exp)^{2}}{exp}=4.2\)

The degrees of freedom of given data

\(=k-1\)

\(=3-1\)

\(=2\)

The critical value, with \(2df\) is \(\chi ^{2}_{0.01}=9.210\)

\(\chi ^{2}=4.2<\chi ^{2}_{0.05}=9.210\)

Because, it does not come in rejection region. So we accept the Null hypothesis.

Hence, the variable has given specified region.