Question

10-65. Consider the hypothesis test H0: σ_1^2=σ_2^2 against H1: σ_1^2<σ_2^2. Suppose that the sample sizes are n1 = 6 and n2 = 13, and that s_1^2=23.8 and s_2^2=27.8. Use α = 0.05. Test the hypothesis and explain how the test could be conducted with a confidence interval on σ_1^2/σ_2^2.

Answer 1

The provided sample variances are
s 23.8
and
85 = 27.8
and the sample sizes are given by and
no13

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: $\sigma_1^2 = \sigma_2^2$ ​

Ha: ​

2 1

This corresponds to a left-tailed test, for which a F-test for two population variances needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the the rejection region for this left-tailed test is R={F:F<FL​=0.214}.

(3) Test Statistics

The F-statistic is computed as follows:

F÷23.8 = 0.856 í 23.8 =-2 =-= 0.856

(4) Decision about the null hypothesis

Since from the sample information we get that F=0.856≥FL​=0.214, 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 the population variance $\sigma_1^2$​ is less than the population variance $\sigma_2^2$​, at the α=0.05 significance level.

Confidence Interval

The 95% confidence interval for
21-22
​​ is: 0.22<​
21-22
<5.586.

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