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Sum
800 people were asked whether they wear glasses for reading with the following results.
Age | Wear glasses | Do not wear glasses |
≤ 30 | 310 | 90 |
> 30 | 290 | 110 |
Compute the χ^{2} square statistic.
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Solution
Table of observed frequencies.
Age | Wear glasses | Do not wear glasses | Row total (R_{i}) |
≤ 30 | 310 | 90 | 60 |
> 30 |
290 | 110 | 400 |
Column total (C_{j}) | 600 | 200 | 800 |
Expected frequencies are given by
E_{ij} = `("R"_"i"xx"C"_"j")/"N"`
E_{11 }= `(400xx600)/800` = 300
E_{12} = `(400xx200)/800` = 100
E_{21 }= `(400xx600)/800` = 300
E_{22} = `(400xx200)/800` = 100
Table of expected frequencies.
Age | Wear glasses | Do not wear glasses | Total |
≤ 30 | 300 | 100 | 400 |
> 30 |
300 | 100 | 400 |
Total | 600 | 200 | 800 |
Now,
χ^{2} = `sum[(("o"_"ij" - "E"_"ij")^2)/"E"_"ij"]`
= `((310 - 300)^2)/(300) + ((90 - 100)^2)/(100) + ((290 - 300)^2)/(300) + ((110 - 100)^2)/(100)`
= `(100)/(300) + (100)/(100) + (100)/(300) + (100)/(100)`
= `200/300+1+1`
= 0.67 + 2
= 2.67
Concept: Chi-Square Statistic ( χ2 )
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