Justify if there is any relationship between sex and color for the following data:

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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.

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