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Unformatted text preview: MN1025 – Business Statistics 39 Lecture 9—Friday 7/3/2008 CHI-SQUARE TEST MOVING AVERAGES Reference: Lind et al. , Chapter 15 for Chi-Square, part of Chapter 19 for Moving Averages. 9.1 Contingency tables These are best understood in the context of an ex- ample: A sales manager is concerned that the brand’s mar- ket share may vary in different parts of the country. To test this, random samples of customers are taken in different parts of the country (North and South) with these results: North South Buy brand 111 69 Do not buy brand 89 31 Is there a bias or not? In other words, is the pattern the same in the North and South? Look at the totals only: North South Total Buy brand 180 Do not buy brand 120 Total 200 100 300 If there were no bias we would expect the 180 in row 1 to divide between North and South in the same proportion as the 120 in row 2. We would therefore expect both the 180 in row 1 and the 120 in row 2 to divide between North and South in the same proportion as the total of 300, that is in the proportion 200:100, or 2:1. So, for row 1 we would expect the figures 180 × 2 3 = 120 and 180 × 1 3 = 60 and, for row 2, 120 × 2 3 = 80 and 120 × 1 3 = 40. A simple way to get these numbers is to multiply the row total by the column total and divide by the overall total. E.g., for row 1, column 1, 180 × 200 300 = 120. To summarise, if there were no bias we would expect the table to look like this: North South Buy brand 120 60 Do not buy brand 80 40 9.2 Chi-Square The numbers observed in our sample are, of course, different from the expected numbers we just calcu- lated. Are they sufficiently different from this pat- tern to imply an underlying difference between the regions? To obtain a measure of the difference be- tween the observed and the expected numbers, we compute Chi-Square as follows. For each cell we take the observed number O , sub- tract the expected number E , square the difference and divide by E to get the Chi-Square contribution for this cell . Then we add these contributions over all cells to get the Chi-Square total χ 2 = summationdisplay ( O- E ) 2 E . The symbol χ is the Greek letter chi. A Chi-Square total which is small or zero suggests no bias (the observed numbers are close to the expected numbers); a large Chi-Square total implies some bias (the difference between the observed and expected numbers is statistically significant). Notice that the Chi-Square total cannot be negative. In our example the first (row 1, column 1) contribu- tion is (111- 120) 2 120 = 0 . 675. The Chi-Square total is obtained by adding the con- tributions for all four cells: (111- 120) 2 120 + (69- 60) 2 60 + (89- 80) 2 80 + (31- 40) 2 40 = 5 . 062 ....
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