Question: Chapter 5 .
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Justify if there is any relationship between sex and color for the following data:

Color Male Female
Red 10 40
White 70 30
Green 30 20
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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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