Question

A manufacturer uses a new production method to produce steel rods. A random sample of 17 steel rods resulted in lengths with a standard deviation of 4.7 cm. At the 0.10 significance level, test the claim that the new production method has lengths with a standard deviation different from 3.5 cm, which was the standard deviation for the old method.

Answer 1

hypothesized variance=3.5²=12.25

sample variance=4.7²=22.09

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:σ²=12.25

Ha:σ²​12.25

This corresponds to a two-tailed test, for which a Chi-Square test for one population variance will be used.

(2) Rejection Region

Based on the information provided, the significance level is \alpha = 0.10α=0.10, and the the rejection region for this two-tailed test is R={χ2:χ2<7.962 or χ2>26.296}.

(3) Test Statistics

The Chi-Squared statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that χ2=28.852>χU2​=26.296, it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population variance σ² is different than 12.25, at the 0.10 significance level.

Hence,σ²​12.25

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