To determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the relative frequency table for the survey results, we need to follow a detailed step-by-step solution:
1. Calculate the Total Number of Respondents:
- First, identify the number of respondents in each category.
- Males who enjoyed the movie: 47
- Males who did not enjoy the movie: 13
- Females who enjoyed the movie: 53
- Females who did not enjoy the movie: 3
The total number of males is:
[tex]\[
\text{Total males} = 47 + 13 = 60
\][/tex]
The total number of females is:
[tex]\[
\text{Total females} = 53 + 3 = 56
\][/tex]
Therefore, the total number of respondents is:
[tex]\[
\text{Total respondents} = 60 + 56 = 116
\][/tex]
2. Calculate the Relative Frequency for Males Who Enjoyed the Movie (a):
- The proportion of males who enjoyed the movie can be calculated as:
[tex]\[
\text{Relative frequency of males enjoyed} = \frac{47}{116} \times 100\%
\][/tex]
Computing this gives:
[tex]\[
\frac{47}{116} \approx 0.4052
\][/tex]
Converting to a percentage:
[tex]\[
0.4052 \times 100 \approx 40.52\%
\][/tex]
Rounding to the nearest whole number:
[tex]\[
a \approx 41\%
\][/tex]
3. Calculate the Relative Frequency for Females Who Did Not Enjoy the Movie (b):
- The proportion of females who did not enjoy the movie is:
[tex]\[
\text{Relative frequency of females did not enjoy} = \frac{3}{116} \times 100\%
\][/tex]
Computing this gives:
[tex]\[
\frac{3}{116} \approx 0.0259
\][/tex]
Converting to a percentage:
[tex]\[
0.0259 \times 100 \approx 2.59\%
\][/tex]
Rounding to the nearest whole number:
[tex]\[
b \approx 3\%
\][/tex]
Thus, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
a = 41\%, \quad b = 3\%
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{a = 41\%,\, b = 3\%}
\][/tex]
