Sum

A sample of men and women who had passed their driving test either in 1^{st} attempt or in 2^{nd} attempt surveyed. Compute χ^{2} statistics.

Passed in → |
First attempt |
Second attempt |

Men |
32 | 28 |

Women |
8 | 12 |

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#### Solution

Table of observed frequencies.

Passed in → |
First attempt |
Second attempt |
Row total (R_{i}) |

Men |
32 | 28 | 60 |

Women |
8 | 12 | 20 |

Column total (C_{j}) |
40 | 40 | 80 |

Expected frequencies are given by

E_{ij} = `("R"_"i"xx"C"_"j")/"N"`

E_{11} = `(60xx40)/80` = 30

E_{12 }= `(60xx40)/80` = 30

E_{21 }= `(20xx40)/80` = 10

E_{22} = `(20xx40)/80` = 10

Table of expected frequencies.

Passed in → |
First attempt |
Second attempt |
Total |

Men |
30 | 30 | 60 |

Women |
10 | 10 | 20 |

Total |
40 | 40 | 80 |

Now,

χ^{2} = `sum[(("O"_"ij" - "E"_"ij")^2)/"E"_"ij"]`

= `((32 - 30)^2)/(30) + ((28 - 30)^2)/(30) + ((8 - 10)^2)/(10) + ((12 - 10)^2)/(10)`

= `4/30+4/30+4/10+4/10`

= `8/30+8/10`

= 0.27 + 0.8

= 1.07

Concept: Chi-Square Statistic ( χ2 )

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