Let's carefully break down the problem and populate the two-way table step-by-step.
1. Total Number of People:
- We know from the problem statement that the total number of survey participants is 50.
2. Total People Who Like and Dislike Cats:
- According to the data, 19 people like cats.
- Therefore, the total number of people who dislike cats is [tex]\( 50 - 19 = 31 \)[/tex].
3. Filling in the Total Column for Dislikes (e):
- We know the total number of people who dislike cats (31).
- Thus, [tex]\( e = 31 \)[/tex].
4. Number of Females Who Dislike Cats:
- We are given that 15 females dislike cats.
5. Number of Males Who Dislike Cats:
- If 31 people dislike cats in total, and 15 of them are females, then the remaining must be males.
- Therefore, the number of males who dislike cats is [tex]\( 31 - 15 = 16 \)[/tex].
6. Total Females (b):
- We need to find out the total number of females.
- We have 15 females who dislike cats and the number of females who like cats [tex]\( a \)[/tex].
- We can use the information that the number of males who like cats (hence, [tex]\( c \)[/tex]) will be [tex]\( 19 - a \)[/tex].
- Summing up all females:
[tex]\[
\text{Total Females} (b) = a + 15
\][/tex]
7. Total Males (d):
- We know the overall population (50) and we know the total number of females (b).
- Therefore, the total number of males will be [tex]\( 50 - b \)[/tex].
8. Females Who Like Cats (a):
- From previous information, we derive [tex]\( a = b - 15 \)[/tex].
Given the detailed data from the problem:
- Final Solution:
[tex]\[
a = 3 \ \text{(females who like cats)} \\
b = 18 \ \text{(total females)} \\
c = 16 \ \text{(males who like cats)} \\
d = 32 \ \text{(total males)} \\
e = 31 \ \text{(total who dislike cats)}
\][/tex]
Therefore, the variables can be filled accordingly in the table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & Likes & Dislikes & Total \\
\hline Female & 3 & 15 & 18 \\
\hline Male & 16 & 16 & 32 \\
\hline Total & 19 & 31 & 50 \\
\hline
\end{tabular}
\][/tex]
