Summary Of A Statistical Test
Essential Eight Communications, Inc. (E8C) is an Oklahoma based Telecommunications Company that currently uses sub-contractors to service and installs their equipment. Some company officers feel it would be more profitable to have full time employees that are able to travel throughout the states of: Arkansas, Louisiana, Oklahoma, and Texas, than it would be to continue to use contractors. This paper will describe the results of an ANOVA test on the West South Central Census Division and Appalachians and West Rockies. It will also show how the use of ANOVA testing can evaluate the solutions of hiring employees to do the same work they contract out.
“A hypothesis is a statement about a population. Data are then used to check the reasonableness of the statement.” (Lind, Marchal & Wathen 2004). The null hypothesis: H0=m1=m2=m3 and the alternate hypothesis: Ha =m1>m2>m3. Where m1 is the mean wage of an E8C contract employee, 27.72, m2 is the mean wage of a telecommunication worker, 19.03, for the West South Central Census Division (National Compensation Survey-Wages), and m3 is the mean wage for the Appalachians and West Rockies. According to Lind, “Hypothesis testing is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.” The level of significance is 95 percent.
“Another use of the F distribution is the analysis of variance (ANOVA) technique in which we compare three or more population means to determine whether they could be equal. To use ANOVA, we assume the following:
1. The populations follow the normal distribution.
2. The populations have equal standard deviations (s).
3. The populations are independent.
When these conditions are met, F is used as the distribution of the test statistic.” (Lind, Marchal & Wathen 2004). ANOVA – The one-way analysis of variance allows us to compare several groups of observations, which are independent but possibly with a different mean for each group. A test of importance is whether or not all the means are equal.
E8C West South Central Division Appalachian’s & West of the Rockies
20.14 17.23 26.27
20.44 20.22 21.74
21.04 12.2 20.78
22.04 17.75 27.31
28.6 15.35 35.65
30.1 18.43 32.45
31.3 19.56 45.25
33.3 21.54 50.12
34.15 25.4 42.53
36.1 22.63 41.62
One factor ANOVA
Mean n Std. Dev
27.04133333 27.721 10 6.2299 E8C
27.04133333 19.031 10 3.7559 West South Central Division
27.04133333 34.372 10 10.2771 Appalachian’s &
West of the Rockies
27.041 30 9.4877 Total
Source SS df MS F p-value
Treatment 1,183.6606 2 591.83030 11.20 .0003
Error 1,426.8325 27 52.84565
Total 2,610.4931 29
Post hoc analysis
Tukey simultaneous comparison t-values (d.f. = 27)
West South Central Division E8C Appalachian’s & West of the Rockies
19.031 27.721 34.372
West South Central Division 19.031
E8C 27.721 2.67
West of the Rockies 34.372 4.72 2.05
critical values for experiment wise error rate:
p-values for pair wise t-tests
West South Central Division E8c Appalachian’s & West of the Rockies
19.031 27.721 34.372
West South Central Division 19.031
E8c 27.721 .0126
West of the Rockies 34.372 .0001 .0506
In accordance with the decision rule if p-value is <(less than) 0.05 then I should reject the null hypothesis H0. I conclude that the wage mean are not equal and that the treatments differ. By following the testing results I reject the null hypothesis and the treatments differ and I found that these treatments are significant. I conclude that it would be cheaper for E8C to continue hiring contractors as laborers throughout the West South Central Division and the Appalachian’s & West of the Rockies than it would be to have full time employees. These findings were based on the wage information contained in the National Compensation Survey: Occupational Wages in the West South Central Census Division, April 2008 which was compiled by the U.S. Department of Labor. The decision has been made to recommend that E8C, Inc hire full-time employees in the effort to make the company more profitable. Describe Test Results The results from the ANOVA testing results reflects that the wage means are not equal and I reject the null hypothesis. E8C average is 27.721 with a variance of 38.81117, West South Central has an average of 19.031 with a variance of 14.10708, Appalachians and West of the Rockies average is 34.372 with a variance of 105.6187 these were all done with the sample count of 10. The source of Variation between groups and within groups is as follows: SOURCE OF VARIATION SS df MS F P-value F crit Between Groups 1183.661 2 591.8303 11.19922 0.000287 3.354131 Within Groups 1426.833 27 52.84565 Total 2610.493 29 The results of the ANOVA test are presented in the ANOVA table shown above. This table contains columns labeled "Source", "SS or Sum of Squares", "df - for degrees of freedom", "MS - for mean square", "F or F-ratio", and "p, prob, probability, sig., or sig. of F". The only columns that are critical for interpretation are the first and the last. The others are used mainly for computational purposes. The row labeled "Between Groups”, having a probability value associated with it, is the only one of any importance at this time. The other rows are used mainly for computational purposes. Of all the information presented in the ANOVA table, the major interest will focused on the value located in the "F crit.” column. If the number (or numbers) found in this column is (are) less than the critical value ( ), then the effect is said to be significant. Since this value is usually set at .05, any value less than this will result in significant effects, while any value greater than this value will result in non-significant effects. If the effects are found to be significant using the above procedure, it implies that the means differ more than would be expected by chance alone. The Within Method Since each of the sample variances may be considered an independent estimate of the parameter , finding the mean of the variances provides a method of combining the separate estimates of into a single value. The resulting statistic is called the MEAN SQUARES WITHIN, often represented by MSW. MSW computes the estimate by combining the variances within each sample. The Between Method The parameter may also be estimated by comparing the means of the different samples, but the logic is slightly less straightforward and employs both the concept of the sampling distribution and the Central Limit Theorem. First, the standard error of the mean squared ( ) is the population variance of a distribution of sample means. In real life scenario, the situation where more than one sample, the variance of the sample means may be used as an estimate of the standard error of the mean squared ( ). In the ANOVA testing, the F-ratio is the statistic used to test the hypothesis that the effects are real: in other words, that the means are significantly different from one another. Therefore, from the findings p-value is <(less than) 0.05 then and I reject the null hypothesis H0. I conclude that the wage mean are not equal and that the treatments differ. References Lind, D.A., Marchal, W.G. and Wathen, S.A. (2004). Statistical Techniques in Business and Economics. The McGraw-Hill Companies, New York. U.S. Department of Labor. National Compensation Survey: Occupational Wages in the West South Central Census Division. Retrieved on April 11, 2008. From: www.bls.gov/ncs/ocs/compub.htm Statistica. StatSoft Inc. 1984-2008. Retrieved on April 11, 2008 From: http://www.statsoft.com/textbook/glosb.html