One hundred people are selected at random and tested for colorblindness to determine whether gender and colorblindness are independent. The following counts were observed.
Colorblind Not Colorblind Total
Male 8 52 60
Female 2 38 40
Total 10 90 100
Find the value of the X2 statistic for the data above.
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To find the value of the X2 statistic for the given data, we need to perform a chi-square test of independence. The formula for calculating the X2 statistic is:
X2 = Σ [(O - E)² / E]
where Σ represents summation, O is the observed frequency, and E is the expected frequency.
First, we need to calculate the expected frequencies for each category by assuming gender and colorblindness are independent. To calculate the expected frequency for a particular cell, use the following formula:
E = (row total * column total) / grand total
Now let's calculate the expected frequencies for each cell:
Colorblind Not Colorblind
Male (60 * 10) / 100 (60 * 90) / 100
6 54
Female (40 * 10) / 100 (40 * 90) / 100
4 36
Next, we calculate (O - E)² / E for each cell:
Colorblind Not Colorblind
Male (8 - 6)² / 6 (52 - 54)² / 54
0.333 0.074
Female (2 - 4)² / 4 (38 - 36)² / 36
0.500 0.222
Now, we sum up all the values: Σ [(O - E)² / E]
(0.333 + 0.074 + 0.500 + 0.222) = 1.129
The X2 statistic for the given data is 1.129.