**Solution:**

Color | Male | Female | Total |
---|---|---|---|

Red | 10 | 40 | 50 |

White | 70 | 30 | 100 |

Green | 30 | 20 | 50 |

Total | 110 | 90 | 200 |

**Step 1:** Null Hypothesis $(H_{0}) \Longrightarrow$ There is no relationship between sex and color.

Alternative Hypothesis $(H_{a}) \Longrightarrow$ There is relationship between sex and color.

**Step 2:** Calculation of test statistic $\Longrightarrow$ On the basis of this hypothesis, the number in the first cell $=\cfrac{A \times B}{N}$

where, A = Number of male in the first column

B = Number of Red color

N = Total number of observation

The number in the first cell$=\cfrac{110 \times 50}{200}=27.5 \approx28$

Similarly fill the table,

Expected frequency table,

Color | Male | Female | Total |
---|---|---|---|

Red | $\cfrac{110 \times 50}{200} = 27.5 \approx 28$ | 50-28=22 (Total-Male) | 50 |

White | $\cfrac{110 \times 100}{200} = 55$ | 100-55=45 (Total-Male) | 100 |

Green | 110-28-55 | 90-22-45=23 | 50 |

Total | 110 | 90 | 200 |

Calculation of $\cfrac{(O-E)^{2}}{E}$

Observed Frequency (O) | Expected Frequency (E) | $(O-E)^{2}$ | $x^{2}=\cfrac{(O-E)^{2}}{E}$ |
---|---|---|---|

10 | 28 | 324 | 11.5714 |

40 | 22 | 324 | 14.7273 |

70 | 55 | 225 | 4.0909 |

30 | 45 | 225 | 5 |

30 | 27 | 9 | 0.3333 |

20 | 23 | 9 | 0.3913 |

Total |
$x^{2}=36.1142$ |

**Step 3:** Level of significance (LOS)

$LOS=0.05=\alpha$

Degree of freedom=(r-1)(c-1)=(2-1)(3-1)=2

**Step 4:** Critical Value $\Longrightarrow$ For 2 degrees of freedom and 5% level of significance, the table value of $(X_{a}^{2})=5.991$

**Step 5:** Decision $\Longrightarrow$

Since the calculated value of $x^{2}=36.1142$ is greater than the table value of $X_{a}^{2}=5.991$.

$\therefore$ The null hypothesis is rejected.

$\therefore$ There is relationship between sex and color.