Unit 2 Section 3 Answers

Agriculture and Environment

Do problem 7.41, page 377. Concentrate on doing the F-test for the difference in population variances. Given the results of this test, perform the associated two independent sample t-test for means? What do you conclude about the distributions of the two groups from these two tests?

Using Minitab we run the STAT > BASIC STATISTICS > 2 VARIANCES procedure to get the following output.
-----------------------------------------
Test for Equal Variances

Level1     Loc_1
Level2     Loc_2
ConfLvl    95.0000
 
Bonferroni confidence intervals for standard deviations
 
  Lower     Sigma     Upper     N  Factor Levels

0.492454  0.752846   1.51597    10  Loc_1
0.258818  0.395671   0.79674    10  Loc_2

 
F-Test (normal distribution)
 

Test Statistic: 3.620
P-Value       : 0.069

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From the above printout, we can see that the value of the F-statistic for comparing the variance of Location 1 (.7528) to the variance of Location 2 (0.39567) is 3.62. This value is the 6.9 percentile of an F distribution with 9 and 9 degrees of freedom. The critical value for said F distribution at the a = 0.025 level is F_a/2,9,9=4.03. Since the computed F is smaller than the critical F we DO NOT reject the null hypothesis and conclude that the two groups have similar variances.

We next run the pooled variances t-test to compare the two location means. Using Minitab again we get:


Two-Sample T-Test and CI: Loc_1, Loc_2

Two-sample T for Loc_1 vs Loc_2

        N      Mean     StDev   SE Mean
Loc_1  10     8.230     0.753      0.24
Loc_2  10     4.090     0.396      0.13

Difference = mu Loc_1 - mu Loc_2
Estimate for difference:  4.140
95% CI for difference: (3.575, 4.705)
T-Test of difference = 0 (vs not =): T-Value = 15.39  P-Value = 0.000  DF = 18
Both use Pooled StDev = 0.601

From this we conclude that the two groups differ significantly in their means (p-value of the test is <0.0005). Hence we REJECT the null hypothesis of equal location means and conclude that the two locations have statistically significant means but common variances. Our best estimate of the parameters for location 1 is mean=8.23 and standard deviation 0.601. The best estimate of the parameters for location 2 is mean=4.09 and standard deviation 0.601.

Think of a situation in your studies or research where testing the equality of variances would provide useful information?