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Sagot :
To determine which two-way frequency table correctly shows the marginal frequencies, let's break down the problem step by step while analyzing the given data and the options provided.
1. Given Data:
- Males who are 6' or over: 12
- Males who are under 6': 86
- Females who are 6' or over: 3
- Females who are under 6': (missing)
First, we need to calculate the missing value for females who are under 6'.
2. Calculate the Total for Females Under 6':
- We should summarize all entries to verify:
Let's assume [tex]\( \text{Number of Females under 6'} = x \)[/tex].
3. Determine the Totals:
- Let's verify the totals provided in the options against our marginal frequencies and ensure we are accurate.
a. Option A Calculation:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & \text{6' or over} & \text{Under 6'} & \text{Total} \\ \hline \text{Male} & 12 & 86 & 98 \\ \hline \text{Female} & 3 & 99 & 102 \\ \hline \text{Total} & 15 & 185 & 200 \\ \hline \end{tabular} \][/tex]
Total males (12 + 86 = 98) matches with given totals.
Total females (3 + 99 = 102) matches with given totals.
Total for 6' or over (12 + 3 = 15) matches with given totals.
Total for under 6' (86 + 99 = 185) matches with given totals.
Grand total (98 + 102 = 200) matches with the overall total.
This matches perfectly with the proper distribution of marginal totals.
b. Option B Calculation:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & \text{6' or over} & \text{Under 6'} & \text{Total} \\ \hline \text{Male} & 12 & 86 & 100 \\ \hline \text{Female} & 3 & 97 & 100 \\ \hline \text{Total} & 15 & 185 & 200 \\ \hline \end{tabular} \][/tex]
Here, Total males (12 + 86 = 98) does not match given total of 100.
Total females (3 + 97 = 100) matches the given total.
So, it is evident that the marginal totals aren’t aligning logically with the individual category totals.
4. Choice: After carefully analyzing and calculating, we find that the marginal frequencies are reflected accurately in Option A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 6' or over & Under 6' & Total \\ \hline Male & 12 & 86 & 98 \\ \hline Female & 3 & 99 & 102 \\ \hline Total & 15 & 185 & 200 \\ \hline \end{tabular} \][/tex]
Thus, the correct option is A.
1. Given Data:
- Males who are 6' or over: 12
- Males who are under 6': 86
- Females who are 6' or over: 3
- Females who are under 6': (missing)
First, we need to calculate the missing value for females who are under 6'.
2. Calculate the Total for Females Under 6':
- We should summarize all entries to verify:
Let's assume [tex]\( \text{Number of Females under 6'} = x \)[/tex].
3. Determine the Totals:
- Let's verify the totals provided in the options against our marginal frequencies and ensure we are accurate.
a. Option A Calculation:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & \text{6' or over} & \text{Under 6'} & \text{Total} \\ \hline \text{Male} & 12 & 86 & 98 \\ \hline \text{Female} & 3 & 99 & 102 \\ \hline \text{Total} & 15 & 185 & 200 \\ \hline \end{tabular} \][/tex]
Total males (12 + 86 = 98) matches with given totals.
Total females (3 + 99 = 102) matches with given totals.
Total for 6' or over (12 + 3 = 15) matches with given totals.
Total for under 6' (86 + 99 = 185) matches with given totals.
Grand total (98 + 102 = 200) matches with the overall total.
This matches perfectly with the proper distribution of marginal totals.
b. Option B Calculation:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & \text{6' or over} & \text{Under 6'} & \text{Total} \\ \hline \text{Male} & 12 & 86 & 100 \\ \hline \text{Female} & 3 & 97 & 100 \\ \hline \text{Total} & 15 & 185 & 200 \\ \hline \end{tabular} \][/tex]
Here, Total males (12 + 86 = 98) does not match given total of 100.
Total females (3 + 97 = 100) matches the given total.
So, it is evident that the marginal totals aren’t aligning logically with the individual category totals.
4. Choice: After carefully analyzing and calculating, we find that the marginal frequencies are reflected accurately in Option A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 6' or over & Under 6' & Total \\ \hline Male & 12 & 86 & 98 \\ \hline Female & 3 & 99 & 102 \\ \hline Total & 15 & 185 & 200 \\ \hline \end{tabular} \][/tex]
Thus, the correct option is A.
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