First, let's establish the data we have and correct any typos for clarity:
We have record counts divided by gender (male and female) and two height categories: "Over 6" and "Under 6." The table provided appears to have an unclear structure, so let’s re-organize the data into a more readable and complete table.
Given data:
- Male, Over 6: 86
- Male, Under 6: 12
- Female, Under 6: 8
Assuming the missing cell is for "Female, Over 6," we need to calculate its value such that the total count remains 200.
### Step-by-Step Solution:
#### Step 1: Define Known Values
Here’s the assumed rectified table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
& \text{Over 6} & \text{Under 6} \\ \hline
\text{Male} & 86 & 12 \\ \hline
\text{Female} & x & 8 \\ \hline
\end{array}
\][/tex]
#### Step 2: Calculate Total Counts for Each Category
From the initial values, we calculate the total number of males and females, and also the totals for each height category.
- Total for Over 6: [tex]\(86\)[/tex] (Male Over 6) [tex]\( + x \)[/tex] (Female Over 6)
- Total for Under 6: [tex]\(12\)[/tex] (Male Under 6) [tex]\( + 8 \)[/tex] (Female Under 6)
- Total males: [tex]\(86 + 12 = 98\)[/tex]
- Total females: [tex]\(x + 8\)[/tex]
#### Step 3: Ensure Consistency with Total Population
The total population is 200, so:
[tex]\[ 98 \text{ (Total Males)} + (x + 8) \text{ (Total Females)} = 200 \][/tex]
This simplifies to:
[tex]\[ x + 106 = 200 \][/tex]
[tex]\[ x = 94 \][/tex]
Thus, the number of females over 6 must be 94.
#### Step 4: Complete the Table
Here is the completed two-way frequency table with marginal frequencies:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& \text{Over 6} & \text{Under 6} & \text{Total} \\ \hline
\text{Male} & 86 & 12 & 98 \\ \hline
\text{Female} & 94 & 8 & 102 \\ \hline
\text{Total} & 180 & 20 & 200 \\ \hline
\end{array}
\][/tex]
#### Verification of Marginal Frequencies:
- Total Over 6: [tex]\(86\)[/tex] (Male) + [tex]\(94\)[/tex] (Female) = [tex]\(180\)[/tex]
- Total Under 6: [tex]\(12\)[/tex] (Male) + [tex]\(8\)[/tex] (Female) = [tex]\(20\)[/tex]
- Total Males: [tex]\(98\)[/tex]
- Total Females: [tex]\(102\)[/tex]
- Population Total: [tex]\(200\)[/tex]
This finalized table and the marginal frequencies correctly show the frequency distribution of the population divided by gender and height categories.
