To solve the given problem, we need to determine the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], [tex]\( d \)[/tex], and [tex]\( e \)[/tex] and fill in the two-way table based on the survey data.
Let's start with the provided values and fill them in step-by-step.
1. Total number of people is 50.
2. Total number of people who like cats is 19 (given).
3. Total number of people who dislike cats is [tex]\( e \)[/tex].
Since the total number of people is divided into those who like and dislike cats, we have:
[tex]\[ e = 50 - 19 = 31 \][/tex]
Step-by-Step Solution:
1. Determine the value of [tex]\( a \)[/tex] (Females who like cats):
We are given that the total number of people who like cats is 19, and we need to distribute this among males and females. Since the total number of males and females together equals 50, and there are 16 males who dislike cats:
Since [tex]\( e = 31 \)[/tex], we distribute the people who like cats:
Total people liking cats (19) - Males who dislike cats (16) = Females who like cats
Thus:
[tex]\[ a = 15 \][/tex]
2. Determine the value of [tex]\( b \)[/tex] (Total number of females):
Total number of females is the sum of females who like cats ([tex]\(a\)[/tex]) and females who dislike cats:
[tex]\[ b = a + 15 \][/tex]
[tex]\[ b = 15 + 15 = 30 \][/tex]
3. Determine the value of [tex]\( c \)[/tex] (Males who like cats):
Since the total number of people who like cats is 19:
[tex]\[ c = 19 - a \][/tex]
[tex]\[ c = 19 - 15 = 4 \][/tex]
4. Determine the value of [tex]\( d \)[/tex] (Total number of males):
Total number of males is the sum of males who like cats ([tex]\(c\)[/tex]) and males who dislike cats:
[tex]\[ d = c + 16 \][/tex]
[tex]\[ d = 4 + 16 = 20 \][/tex]
5. Value of [tex]\( e \)[/tex] is already determined as:
[tex]\[ e = 31 \][/tex]
Now, let's fill in the table with these values:
[tex]\[
\begin{aligned}
a & = 15 \\
b & = 30 \\
c & = 4
\end{aligned}
\][/tex]
[tex]\[
\begin{tabular}{|c|c|c|c|}
\cline{2-4}
& Likes & Dislikes & Total \\
\hline Female & 15 & 15 & 30 \\
\hline Male & 4 & 16 & 20 \\
\hline Total & 19 & 31 & 50 \\
\hline
\end{tabular}
\][/tex]
Thus, the determined values are:
[tex]\[
d = 20 \quad e = 31
\][/tex]
