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A group of 65 baseball players were surveyed about which hand they favor for batting. The data from the survey are shown in the Venn diagram.

Determine the value for each variable in the two-way table.

[tex]\[
\begin{array}{l}
a= \\
b= \\
c=
\end{array}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & Right & Left & Total \\
\hline
Female & 24 & $a$ & $b$ \\
\hline
Male & $c$ & 6 & 34 \\
\hline
Total & 52 & $d$ & $e$ \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{array}{l}
d= \\
e=
\end{array}
\][/tex]



Answer :

GinnyAnswer
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]

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