Question

A research hypothesis is that the variance of stopping distances of automobiles on wet pavement is significantly greater than the variance of stopping distances of automobiles on dry pavement. In the research study, 21 automobiles traveling at the same speeds are tested for stopping distances on wet pavement and then tested for stopping distances on dry pavement. On wet pavement the standard deviation of stopping distances is 32 feet. On dry pavement, the standard deviation is 16 feet. If the hypothesis test with  is to be run with a significance level of 1%, then provide the value of the critical number that you would use to set up the Rejection Region.

Answer 1

The given data are summarized as follows:

	Sample 1	Sample 2
Sample size	n1=21	n2=21
Sample Standard Deviation	=32	=16

Given the significance level is as 1% i.e. =0.01

As the population mean and the population SD are unknown, it needs to be estimated by the sample mean and the sample SD and hence the critical value is obtained from the Biometrika table as follows:

You have only wanted the critical value , but I am doing the whole test for better explanation. Ignore if you dont need it.

:Let the population variance of the first population be and the population variance of the second population be . The null and the alternate hypotheses are given by

Take,

Now, the hypotheses can be written as

The test statistic,under the null hypothesis, is given by

The test statistic under the null hypothesis, follows F distribution with df 20,20.

As the observed value is greater than the critical value, we reject the null hypothesis at 1% level of significance and conclude that the variance of stopping distances of automobiles on wet pavement is significantly greater than the variance of the stopping distances of automobiles on dry pavement.

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