Ch15-1 - Chapter 15 Analysis of Variance 15.1 Introduction...

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Analysis of Variance Chapter 15
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15.1 Introduction Analysis of variance compares two or more populations of interval data. Specifically, we are interested in determining whether differences exist between the population means. The procedure works by analyzing the sample variance.
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The analysis of variance is a procedure that tests to determine whether differences exits between two or more population means. To do this, the technique analyzes the sample variances 15.2 One Way Analysis of Variance
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Example 15.1 An apple juice manufacturer is planning to develop a new product -a liquid concentrate. The marketing manager has to decide how to market the new product. Three strategies are considered Emphasize convenience of using the product. Emphasize the quality of the product. Emphasize the product’s low price. One Way Analysis of Variance
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Example 15.1 - continued An experiment was conducted as follows: In three cities an advertisement campaign was launched . In each city only one of the three characteristics (convenience, quality, and price) was emphasized. The weekly sales were recorded for twenty weeks following the beginning of the One Way Analysis of Variance
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One Way Analysis of Variance Convnce Quality Price 529 804 672 658 630 531 793 774 443 514 717 596 663 679 602 719 604 502 711 620 659 606 697 689 461 706 675 615 512 498 492 691 733 787 698 495 699 776 485 572 561 557 523 353 584 469 634 581 542 580 614 624 532 See file Xm15 -01 Weekly sales Week ly sales Weekl y sales
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Solution The data are interval The problem objective is to compare sales in three cities. We hypothesize that the three population means are equal One Way Analysis of Variance
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H 0 : μ 1 = μ 2 = μ 3 H 1 : At least two means differ To build the statistic needed to test the hypotheses use the following notation: Solution Defining the Hypotheses
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ependent samples are drawn from k populations (treatme 1 2 k X 11 x 21 . . . X n1,1 1 n X 12 x 22 . . . X n2,2 2 2 x n X 1k x 2k . . . X nk,k k k x n Sample mean First observation, first sample Second observation, second sample X is the “response variable”. The variables’ value are called “responses”. Notation
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Terminology In the context of this problem… Response variable – weekly sales Responses – actual sale values Experimental unit – weeks in the three cities when we record sales figures. Factor – the criterion by which we classify the populations (the treatments). In this problems the factor is the marketing strategy. Factor levels – the population (treatment) names. In this problem factor levels are the marketing trategies.
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Two types of variability are employed when testing for the equality of the population means The rationale of the test statistic
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Graphical demonstration: Employing two types of variability
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20 25 30 1 7 Treatment 1 Treatment 2 Treatment 3 10 12 19 9 Treatment 1 Treatment 2 Treatment 3 20 16 15 14 11 10 9 10 x 1 = 15 x 2 = 20 x 3 = 10 x 1 = 15 x 2 = 20 x 3 = The sample means are the same as before, but the larger within-sample variability makes it harder to draw a conclusion about the population means.
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Ch15-1 - Chapter 15 Analysis of Variance 15.1 Introduction...

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