To determine the values for the variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], [tex]\(d\)[/tex], and [tex]\(e\)[/tex] in the two-way table, let’s analyze the given question step by step.
We know the total number of surveyed baseball players is:
[tex]\[ e = 65 \][/tex]
### Given Information
- Total surveyed players: [tex]\( e = 65 \)[/tex]
- Total players who favor their right hand for batting: 52
- Total males surveyed: 34
- Total males who favor their left hand: 6
- Total females who favor their right hand: 24
### Finding [tex]\(a\)[/tex] - Number of Females who favor the Left Hand
First, we need to find the total number of players who favor their left hand:
[tex]\[ d = \text{Total surveyed} - \text{Total who favor right hand} \][/tex]
[tex]\[ d = 65 - 52 \][/tex]
[tex]\[ d = 13 \][/tex]
Next, we will calculate the number of females who favor the left hand:
[tex]\[ a = d - (\text{Number of males who favor the left hand}) \][/tex]
[tex]\[ a = 13 - 6 \][/tex]
[tex]\[ a = 7 \][/tex]
### Finding [tex]\(b\)[/tex] - Total Number of Female Players
To find the total number of female players:
[tex]\[ b = e - (\text{Total males surveyed}) \][/tex]
[tex]\[ b = 65 - 34 \][/tex]
[tex]\[ b = 31 \][/tex]
### Finding [tex]\(c\)[/tex] - Number of Males who favor the Right Hand
To find the number of males who favor the right hand:
[tex]\[ c = \text{Total who favor right hand} - (\text{Number of females who favor right hand}) \][/tex]
[tex]\[ c = 52 - 24 \][/tex]
[tex]\[ c = 28 \][/tex]
### Summarizing the Missing Values
Now we have all the values:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 31 \)[/tex]
- [tex]\( c = 28 \)[/tex]
- [tex]\( d = 13 \)[/tex]
- [tex]\( e = 65 \)[/tex]
So the final two-way table would look like this:
[tex]\[
\begin{array}{l}
a= 7 \\
b= 31 \\
c= 28
\end{array}
\][/tex]
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & \text{Right} & \text{Left} & \text{Total} \\
\hline \text{Female} & 24 & 7 & 31 \\
\hline \text{Male} & 28 & 6 & 34 \\
\hline \text{Total} & 52 & 13 & 65 \\
\hline
\end{array}
\][/tex]
