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Ch16 - Chapter 16 Chi Squared Tests 16.1 Introduction Two...

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Chi Squared Tests Chapter 16
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16.1 Introduction Two statistical techniques are presented, to analyze nominal data. A goodness-of-fit test for the multinomial experiment. A contingency table test of independence. Both tests use the χ 2 as the sampling distribution of the test statistic.
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The hypothesis tested involves the probabilities p 1 , p 2 , …, p k. of a multinomial distribution. The multinomial experiment is an extension of the binomial experiment. There are n independent trials. The outcome of each trial can be classified into one of k categories, called cells. The probability p i that the outcome fall into cell i remains constant for each trial. Moreover, p 1 + p 2 + … +p k = 1. Trials of the experiment are independent 16.2 Chi-Squared Goodness-of- Fit Test
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16.2 Chi-squared Goodness-of- Fit Test We test whether there is sufficient evidence to reject a pre-specified set of values for p i. The hypothesis: i i k k a p one least At H a p a p a p H = = = : ,..., , : 1 2 2 1 1 0 The test builds on comparing actual frequency and the expected frequency of occurrences in all the cells.
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Example 16.1 Two competing companies A and B have enjoy dominant position in the market. The companies conducted aggressive advertising campaigns. Market shares before the campaigns were: Company A = 45% Company B = 40% Other competitors = 15%. The multinomial goodness of fit test - Example
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Example 16.1 continued To study the effect of the campaign on the market shares, a survey was conducted. The multinomial goodness of fit test - Example 200 customers were asked to indicate their preference regarding the product advertised. Survey results: 102 customers preferred the company A’s product, 82 customers preferred the company B’s product,
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The multinomial goodness of fit test - Example Example 16.1 continued Can we conclude at 5% significance level that the market shares were affected by the advertising campaigns?
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Solution The population investigated is the brand preferences. The data are nominal (A, B, or other) This is a multinomial experiment (three categories).
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