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 0.83. 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 placebo group. Assume that the two populations are normally distributed. What are the null and alternative​ hypotheses? A. Ho​: LaTeX: \sigma_1^2 σ 1 2 ≠ LaTeX: \sigma_2^2 σ 2 2 H1​:LaTeX: \sigma_1^2 σ 1 2 = LaTeX: \sigma_2^2 σ 2 2 B. Ho​: LaTeX: \sigma_1^2 σ 1 2 = LaTeX: \sigma_2^2 σ 2 2 H1: LaTeX: \sigma_1^2 σ 1 2 > LaTeX: \sigma_2^2 σ 2 2 C. Ho​: LaTeX: \sigma_1^2 σ 1 2 = LaTeX: \sigma_2^2 σ 2 2 H1​: LaTeX: \sigma_1^2 σ 1 2 < LaTeX: \sigma_2^2 σ 2 2 D. Ho​: LaTeX: \sigma_1^2 σ 1 2 = LaTeX: \sigma_2^2 σ 2 2 H1: LaTeX: \sigma_1^2 σ 1 2 ≠ LaTeX: \sigma_2^2 σ 2 2 Identify the test statistic. ________________​ (Round to two decimal places as​ needed.) Use technology to identify the​ P-value. ______________ ​(Round to three decimal places as​ needed.) What is the conclusion for this hypothesis​ test? A. Fail to reject Ho. 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. B. Reject Ho. There is insufficient evidence to support the claim that the treatment group has errors that vary significantly more than the errors of the placebo group. C. Reject Ho. 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. D. Fail to reject Ho. There is insufficient evidence to support the claim that the treatment group has errors that vary significantly more than the errors of the placebo group.

Answer 1

We want to test the claim that the treatment group has errors that vary significantly more than the errors of the placebo group.

To test this claim the Null and alternative Hypothesis:

Against

Where,

=Population variance of the errors for the treatment group

=population variance of the errors for the placebo group

Test statistic:

To test equality of two population vaiances the F- test statistic is:

Is follow F Distribution with degrees of freedom.  

s_{​​​​​​1}= sample standard deviation of the errors for the treatment group= 2.40​

= sample variance of the errors for the treatment group= 2.40​²

= sample variance of the errors for the placebo group=0.83²

n_{​​​​​​1}= sample size of treatment group peoples=21

n_{​​​​​2}= sample size of placebo group peoples=21

F=8.3612

The critical value fot F test at = (21-1),(21-1)=20,20 degrees of freedom is:

=2.124

P-value= 0.00001 < 0.05 And

Cal F > critical value of F

Conclusion:

Reject H_{​​​​​​0} at 5 % level of significance.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.