Question

researchers conducted an experiment to test the effects of alcohol. Errors were recorded in a test of visual and motor skills for a treatment group of 21 people who drank ethanol and another group of 21 people given a placebo. The errors for the treatment group have a standard deviation of 2.40 and the errors for the placebo group; have a standard deviation of 0.73 use a 0.05 significance level to test the claim that the treatment group has errors that vary significantly more than the errors of the placeo group assume that the two populations are normally distributed.

what is the null and alternative hypotheses?

what is the test statistic?

what is the p value

what is the conclusion?

Answer 1

Solution :

The null and alternative hypotheses are as follows :

Where, is population variance of errors of treatment group and is population variance of errors of placebo group.

To test the hypothesis we shall use F test for testing the equality of two population variances. The test statistic is given as follows :

Where, are sample standard deviations for treatment groups and placebo groups respectively.

The test statistic follows F distribution with (n​​​​​​1 - 1, n​​​​​​2 - 1) degrees of freedom.

We have,

The value of the test statistic is 10.8088.

The degrees of freedom = (21 - 1, 21 - 1) = (20, 20)

Since, our test is right-tailed test, therefore we shall obtain right-tailed p-value for the test statistic. The right-tailed p-value is given as follows :

p-value = P(F > value of the test statistic)

p-value = P(F > 10.8088)

p-value = 0.0000

The p-value is 0.0000.

Decision :

Significance level = 0.05

p-value = 0.0000

(0.0000 < 0.05)

Since, p-value is less than the significance level of 0.05, therefore we shall reject the null hypothesis (H​​​​​​0) at 0.05 significance level.

Conclusion : At significance level of 0.05, there is sufficient evidence to support the claim that the treatment group has errors that vary significantly more than the errors of the placebo group.

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