Question

A distribution and the observed frequencies of the values of a variable from a simple random sample of the population are provided below. 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: 0.3​, 0.2​, 0.2​, 0.2​, 0.1  

Observed​ frequencies: 12​, 9​, 8​, 17​, 4

Significance level = 0.10

Answer 1

From the given information,

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=0.3,p2​=0.2,p3​=0.2,p4​=0.2,p5​=0.1 or distribution of variable is same as the given distribution.

Ha: Some of the population proportions differ from the values stated in the null hypothesis

This corresponds to a Chi-Square test for Goodness of Fit.

(2) Rejection Region

Based on the information provided, the significance level is α=0.10, the number of degrees of freedom is df = 5 - 1 = 4, so then the rejection region for this test is R = {χ2:χ2>7.779}.

(3) Test Statistics

The Chi-Squared statistic is computed as follows:

chi^2 = 6.2

(4) Decision about the null hypothesis

Since it is observed that χ2=6.2≤ 7.779, it is then concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at α=0.10 significance level.

Hence,

Distribution of variable is same as of given distribution.

Thank you.