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QSO 510 Module Seven 1

Module Six introduced hypotheses and hypothesis testing on a single population mean. Module Seven compares several population means through a statistical procedure called analysis of variance (ANOVA). One-way ANOVA, also referred to as one-factor ANOVA or completely randomized design, is a part of Design of Experiments, a larger subset of statistics used extensively in the automotive, chemical, and medicinal drug industries. How does the ANOVA test work? To determine whether the various sample means came from a single population or populations with different means, you actually compare these sample means through their variances. For example, a general manager of a chemical plant may wish to determine whether a difference exists in the annual salaries of his shift supervisors, assistant plant managers, and maintenance managers. Within-group variation exists among salaries in each of the three groups, and between-group variation is present across the three groups. ANOVA uses a ratio of between-group variation to within-group variation to form an F statistic. If the F statistic results in a p value that is less than or equal to a given significance level (typically 5%), then he may conclude that the salaries of shift supervisors, assistant plant managers, and maintenance managers are significantly different. If the p value exceeds the significance level, then the annual salaries of the three groups are not significantly different. Note that probability computation for an F statistic is based on an F distribution. There is not a single F distribution but a family of F distributions. A particular member of the family is determined by two parameters: the degrees of freedom in the numerator and the degrees of freedom in the denominator. Consider another example of ANOVA. A professor taught four small sections of Quantitative Analysis last semester, which resulted in the following data on student scores by section:

Section 1 Section 2 Section 3 Section 4

94 75 70 68

90 68 73 70

85 77 76 72

80 83 78 65

88 80 74

68 65

65

2 QSO 510 Module Seven

The professor would like to know whether there is a difference in the mean scores for students in the four sections. Using statistical software to analyze the data with ANOVA provides the following results:

ANOVA Source of

Variation SS df MS F P value F crit

Between Groups 440.4933 3 146.8311 2.530122 0.089662 3.159908 Within Groups 1044.598 18 58.0332

Total 1485.091 21

Note that F = 2.53 and p = 0.089662. At a significance level of 0.05, H0 will not be rejected and we conclude that the mean scores of students in the four sections of the course are not significantly different. Additional applications of ANOVA may include a researcher using ANOVA to test for a difference in the effectiveness of three drugs in treating Alzheimer’s disease. Or, an automotive engineer may use ANOVA to test for a difference in three fuel blends on the performance of the company’s new engine. An operations manager may use ANOVA to test for a difference in delivery times for the company’s products over four routes.