Questions: In order to determine if blue eye color and blonde hair color differ significantly, a chi-square test for homogeneity should be performed. What is the expected frequency of blonde hair and blue eyes (answer choices are rounded to the nearest hundredth)? a) 27.67 b) 25.50 c) 28.41 d) 27.25

In order to determine if blue eye color and blonde hair color differ significantly, a chi-square test for homogeneity should be performed.

What is the expected frequency of blonde hair and blue eyes (answer choices are rounded to the nearest hundredth)?
a) 27.67
b) 25.50
c) 28.41
d) 27.25

Transcript text: In order to determine if blue eye color and blonde hair color differ significantly, a chi-square test for homogeneity should be performed. What is the expected frequency of blonde hair and blue eyes (answer choices are rounded to the nearest hundredth)? a) 27.67 b) 25.50 c) 28.41 d) 27.25

Solution

Solution Steps

Step 1: Calculate Row and Column Totals

To find the expected frequency of blonde hair and blue eyes, we first calculate the row totals and column totals from the observed frequencies in the contingency table.

The observed frequencies are: \[ \begin{array}{|l|l|l|l|} \hline & \text{Blue} & \text{Green} & \text{Brown} \\ \hline \text{Blonde} & 25 & 27 & 31 \\ \hline \text{Brown} & 26 & 18 & 22 \\ \hline \end{array} \]

Calculating the row totals:

For Blonde: \( 25 + 27 + 31 = 83 \)
For Brown: \( 26 + 18 + 22 = 66 \)

Calculating the column totals:

For Blue: \( 25 + 26 = 51 \)
For Green: \( 27 + 18 = 45 \)
For Brown: \( 31 + 22 = 53 \)

The grand total is: \[ 83 + 66 = 149 \]

Step 2: Calculate Expected Frequency

The expected frequency \( E \) for blonde hair and blue eyes is calculated using the formula: \[ E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}} \]

Substituting the values: \[ E = \frac{(83 \times 51)}{149} \]

Calculating this gives: \[ E \approx 28.41 \]