Question

1. (NOTE: This exercise is the one exception to the no-by-hand-work rule. Here, you must type in answers to all parts of the exercise.) We wish to compare the variances of two populations. We take random samples from each with the following results:

Sample 1: n = 38, sample variance = 32.4

Sample 2: n = 44, sample variance = 17.6

We wish to use the F test to test whether the population variances are equal.

1. What are the hypotheses for a 1-sided F test?
2. What is the F statistic for this test?
3. What are the degrees of freedom (numerator and denominator) for this test?
4. The critical value for a 1-sided F test based on these samples at α = 0.05 is 1.685. Given this and assuming a valid test, do the test by hand and state the conclusion of this test.
5. Now, use Minitab to conduct the test. (Use 2-Variances with Summarized Data.) Interpret the corresponding ratio confidence bound for this test.

Answer 1

a) We will set the null hypothesis

H₀ : VS   H1 :

b) The F.test statistics is

we are given that S1²=32.4, n1=38,   S2²=17.6 and n2=44,

c) The degree of freedom for numerator and denominator is

DF(Numerator)	DF2(deniminator)
37	43

d) We are given the critical value for f test at 0.05 level of significance = 1.685

The calculated f test is given below

Method	DF(Numerator)	DF2(deniminator)	F.Statistic	P-Value
F Test (normal)	37	43	1.84	0.055

Since the calculated F ratio =1.84 is greater then tabulated F.Ratio =1.685 we accept our null hypothesis and conclude that the two variance are equal. or in other words since calculated P.value=0.055 is greater then 0.05 we accept our null and reject our alternate hypothesis at 0.05 level of significance.

e) MiniTab Output
Test and CI for Two Variances

Method

Null hypothesis Sigma(1) / Sigma(2) = 1
Alternative hypothesis Sigma(1) / Sigma(2) not = 1
Significance level Alpha = 0.05

Statistics

Sample N StDev Variance
1 38 5.692 32.400
2 44 4.195 17.600

Ratio of standard deviations = 1.357
Ratio of variances = 1.841

95% Confidence Intervals

CI for
Distribution CI for StDev Variance
of Data Ratio Ratio
Normal (0.994, 1.867) (0.987, 3.485)

Tests

Test
Method DF1 DF2 Statistic P-Value
F Test (normal) 37 43 1.84 0.055