Alright, let's solve this step-by-step.
### Step-by-Step Solution
1. Understanding the Given Data:
- The table represents the number of caries, the frequency ([tex]\( f_i \)[/tex]), and the relative frequency ([tex]\( h_i \)[/tex]).
- Let's rewrite the table with the missing values indicated:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Number of Caries} & f_i & h_i \\
\hline
0 & 25 & 0.25 \\
\hline
1 & 20 & 0.20 \\
\hline
2 & a & c \\
\hline
3 & 15 & 0.15 \\
\hline
4 & b & 0.05 \\
\hline
\text{Total} & & \\
\hline
\end{array}
\][/tex]
2. Sum of Relative Frequencies:
- The sum of all relative frequencies should be 1.
- Given relative frequencies so far are:
[tex]\[
0.25 + 0.20 + c + 0.15 + 0.05 = 1
\][/tex]
- Let's find [tex]\( c \)[/tex]:
[tex]\[
c = 1 - (0.25 + 0.20 + 0.15 + 0.05) = 1 - 0.65 = 0.35
\][/tex]
3. Total Sum of Frequencies:
- To find [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we need to know the total sum of frequencies.
- Given frequencies so far are:
[tex]\[
25 + 20 + a + 15 + b = \text{Total Frequency}
\][/tex]
- Since [tex]\( h_i = \frac{f_i}{\text{Total Frequency}} \)[/tex]:
[tex]\[
0.25 = \frac{25}{\text{Total Frequency}}
\][/tex]
- Hence, Total Frequency = 100.
4. Calculate Missing Frequencies:
- Now, let's find [tex]\( f_i \)[/tex] for [tex]\( a \)[/tex] and [tex]\( b \)[/tex], knowing [tex]\( h_i \)[/tex]:
- For [tex]\( a \)[/tex]:
[tex]\[
h_i = 0.35 \implies a = h_i \times \text{Total Frequency} \implies a = 0.35 \times 100 = 35
\][/tex]
- For [tex]\( b \)[/tex]:
[tex]\[
h_i = 0.05 \implies b = h_i \times \text{Total Frequency} \implies b = 0.05 \times 100 = 5
\][/tex]
5. Assertion for Validation:
- Ensure:
[tex]\[
\text{Sum of frequencies} = 25 + 20 + 35 + 15 + 5 = 100
\][/tex]
- And:
[tex]\[
\text{Sum of relative frequencies} = 0.25 + 0.20 + 0.35 + 0.15 + 0.05 = 1
\][/tex]
Based on the above solution, the missing values are:
- [tex]\( a = 35 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- Total Frequency = 100
