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A group of 50 people were asked their gender and if they liked cats. The data from the survey are shown in the Venn diagram. Determine the value for each variable in the two-way table.

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

\begin{tabular}{|c|c|c|c|}
\hline & Likes & Dislikes & Total \\
\hline Female & [tex]$a$[/tex] & 15 & [tex]$b$[/tex] \\
\hline Male & [tex]$c$[/tex] & 16 & [tex]$d$[/tex] \\
\hline Total & 19 & [tex]$e$[/tex] & 50 \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
Sure, let's go through the solution step by step.

Given the table:
[tex]\[ \begin{array}{l} a=\square \\ b=\square \\ c=\square \end{array} \][/tex]
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & Likes & Dislikes & Total \\
\hline Female & [tex]$a$[/tex] & 15 & [tex]$b$[/tex] \\
\hline Male & [tex]$c$[/tex] & 16 & [tex]$d$[/tex] \\
\hline Total & 19 & [tex]$e$[/tex] & 50 \\
\hline
\end{tabular}

Let's break down how to find each of these variables.

1. Calculate [tex]\(a\)[/tex] (Number of females who like cats):

- The total number of likes is 19.
- We know some of these likes are from males and the rest are from females.
- We also need to find the number of males who like cats ([tex]\(c\)[/tex]).

Initially, if we don't know [tex]\(c\)[/tex] yet but we know that [tex]\(a + c = 19\)[/tex].
Given [tex]\(c = 4\)[/tex] (we'll confirm this in a moment), solving for [tex]\(a\)[/tex] gives us [tex]\(a = 19 - c = 19 - 4 = 15\)[/tex].

So,
[tex]\[ a = 15 \][/tex]

2. Calculate [tex]\(b\)[/tex] (Total number of females):

- This is the sum of females who like and dislike cats.
- We have [tex]\(a = 15\)[/tex] and the number of females who dislike cats is 15.

Thus,
[tex]\[ b = a + 15 = 15 + 15 = 30 \][/tex]

3. Calculate [tex]\(c\)[/tex] (Number of males who like cats):

- Given the earlier calculation and confirmed data, [tex]\(c = 4\)[/tex].

So,
[tex]\[ c = 4 \][/tex]

4. Calculate [tex]\(d\)[/tex] (Total number of males):

- This is the sum of males who like and dislike cats.
- We have [tex]\(c = 4\)[/tex] and the number of males who dislike cats is 16.

Thus,
[tex]\[ d = c + 16 = 4 + 16 = 20 \][/tex]

5. Calculate [tex]\(e\)[/tex] (Total number of people who dislike cats):

- Total number of dislikes should equal the total number of people minus the total number of likes.
- Total number of people is 50 and total likes is 19.

Thus,
[tex]\[ e = 50 - 19 = 31 \][/tex]

Now, let's fill in the table with these values:

[tex]\[ \begin{array}{l} a=15 \\ b=30 \\ c=4 \end{array} \][/tex]
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & Likes & Dislikes & Total \\
\hline Female & 15 & 15 & 30 \\
\hline Male & 4 & 16 & 20 \\
\hline Total & 19 & 31 & 50 \\
\hline
\end{tabular}

So, the complete solution is:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 30\)[/tex]
- [tex]\(c = 4\)[/tex]
- [tex]\(d = 20\)[/tex]
- [tex]\(e = 31\)[/tex]
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