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 7 53 60
Female 1 39 40
Total 8 92 100
Find the value of the X2 statistic for the data above.
A. 1.325
B. 1.318
C. 1.286
D. 1.264
Not getting anything in that range..
To find the value of the Chi-square (X^2) statistic for the given data, we can use the formula:
X^2 = Σ ( (O - E)^2 / E )
where Σ represents the summation, O is the observed frequency, and E is the expected frequency.
To calculate the expected frequency, we need to assume that gender and colorblindness are independent. Given that there are 100 people in total, we can calculate the expected frequency for each cell using the formula:
E = (row total * column total) / grand total
Now, let's calculate the expected values and the Chi-square statistic step by step:
First, let's calculate the expected frequency for each cell:
Expected frequency for Male and Colorblind cell:
E(Male, Colorblind) = (total Male * total Colorblind) / grand total
= (60 * 8) / 100
= 4.8
Expected frequency for Male and Not Colorblind cell:
E(Male, Not Colorblind) = (total Male * total Not Colorblind) / grand total
= (60 * 92) / 100
= 55.2
Expected frequency for Female and Colorblind cell:
E(Female, Colorblind) = (total Female * total Colorblind) / grand total
= (40 * 8) / 100
= 3.2
Expected frequency for Female and Not Colorblind cell:
E(Female, Not Colorblind) = (total Female * total Not Colorblind) / grand total
= (40 * 92) / 100
= 36.8
Now, let's calculate the Chi-square statistic:
X^2 = Σ ( (O - E)^2 / E )
Using the observed (O) and expected (E) frequencies from the table, we get:
X^2 = ( (7 - 4.8)^2 / 4.8 ) + ( (53 - 55.2)^2 / 55.2 ) + ( (1 - 3.2)^2 / 3.2 ) + ( (39 - 36.8)^2 / 36.8 )
Simplifying the equation:
X^2 = ( 4.41 / 4.8 ) + ( 4.84 / 55.2 ) + ( 2.25 / 3.2 ) + ( 2.25 / 36.8 )
= 0.91875 + 0.08768 + 0.70313 + 0.06114
= 1.7707
Therefore, the value of the X^2 statistic for the given data is approximately 1.7707.
Since none of the given answer choices match this value, there might be a calculation error or discrepancy in the data provided.