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Illustration of Chi-Square tests A wonderful journey to the ABC Fruitcake Factory.

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Goodness of Fit Are the proportions of the ingredients the same throughout the sample? Do the proportions of ingredients conform to a predetermined recipe?

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Properties of Goodness of Fit Expected cell counts should be at least five For a sample of k distinct observations the Degrees of Freedom are (k-1). Tests to determine if the proportions of possible observations are equivalent or equal to a predetermined demographic split of the population.

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Example ABC Fruitcake prefers that the proportion of red widgets, green widgets, and crunchy widgets should be.25,.45, and.3 respectively. In a sample of fruitcake, we observe the following frequency of widgets: Widget typeFrequency Red236 Green443 Crunchy321 Is ABC Fruitcake achieving the correct proportion of ingredients in their fruitcakes?

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Test Of Independence Does the proportion of a certain ingredient influence the proportion of another?

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Properties of Independence Fixed overall N ( a single large sample of sufficient size such that the expected count of the cells is at least 5). Sample is assumed to be randomly selected. Tests to determine if a given Row factor is dependent on a given column factor. For an i X j table the degrees of freedom are (i-1)(j-1). Columns can be combined if the expected counts in any cell is less than 5

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Example ABC Fruitcake is interested in the the relationship of the level of crunchy widgets and the proportion of green widgets. Crunchy level Green Widgets Red widgets Marginal Totals Low563995 High7044114 Marginal totals 12683209 Management would like to know if the level of crunchy widgets is associated with the proportion of green widgets in the fruitcake.

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Test of Homogeneity Marginal totals fixed Do the proportions of ingredients vary between different brands?

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Properties of Homogeneity Marginal totals fixed (The rows of the table often denote the separate samples under question). Tests to determine if the populations of the different samples have the same characteristics. Samples are assumed to be independent and randomly selected. For an i X j table the degrees of freedom are (i-1)(j-1) Columns can be combined if the expected counts in any cells are less than 5.

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Example Fruitcakes come in two varieties, one which has a low level of crunchy widgets and one which has a high level of crunchy widgets. ABC Fruitcakes management would like to know if the proportions of red and green widgets are the same between the different types of fruitcake. Crunchy level Green Widgets Red widgets Marginal Totals Low563995 High7044114 Marginal totals 12683209

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So what is the difference using Minitab? The tests are both performed using the same Minitab Commands, the difference falls in the hypotheses. The interpretation of the results also changes.

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Sources: The Fruitcake Lady. January 16, 2006. IndexHere January 18, 2007. http://indexhere.blogspot.com/2006/01/fruitcake- lady.html Data Analysis 2 course Hot sheet. Rochester Institute of Technology. January 18, 2007.

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McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Chi-Square Tests Chapter 12.

McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Chi-Square Tests Chapter 12.

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