Question

A researcher wants to study the relationship between salary and gender. She randomly selects 316individuals and determines their salary and gender. Can the researcher conclude that salary and gender are dependent?

Income | Male | Female | Total |
---|---|---|---|

Below $25,000$ | 36 | 68 | 104 |

$25,000-$50,000 | 54 | 43 | 97 |

$50,000-$75,000 | 42 | 31 | 73 |

Above $75,000 | 23 | 19 | 42 |

Total | 155 | 161 | 316 |

-Find the critical value of the test at the 0.01 level of significance. Round your answer to three decimal places.

-Make the decision to reject or fail to reject the null hypothesis at the 0.01 level of significance?

-State the conclusion of the hypothesis test at the 0.01 level of significance? (Sufficient evidence or not sufficient evidence?)

Answer #1

Applying chi square test of independence: |

Expected |
E_{i}=row total*column
total/grand total |
male | female | Total |

<25000 | 51.01 | 52.99 | 104 | |

25000-50000 | 47.58 | 49.42 | 97 | |

50000-75000 | 35.81 | 37.19 | 73 | |

>75000 | 20.60 | 21.40 | 42 | |

total | 155 | 161 | 316 | |

chi square χ^{2} |
=(O_{i}-E_{i})^{2}/E_{i} |
male | female | Total |

<25000 | 4.4181 | 4.2535 | 8.6716 | |

25000-50000 | 0.8665 | 0.8342 | 1.7007 | |

50000-75000 | 1.0711 | 1.0312 | 2.1023 | |

>75000 | 0.2793 | 0.2689 | 0.5482 | |

total | 6.6351 | 6.3878 | 13.0228 | |

test statistic X^{2} = |
13.023 |

for 3 df and 0.01 level , critical
value χ2= |
11.345 |

reject the null hypothesis at the 0.01 level of significance

Sufficient evidence

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