Certainly, let's go step-by-step to solve the given problem.
We are provided a frequency distribution table for masses of dogs and need to determine the number of dogs that have a mass greater than 24 kg.
Let's look at the data first:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Mass, } x (\text{kg}) & \text{Frequency} \\
\hline
0 \leq x < 10 & 2 \\
\hline
10 \leq x < 20 & 7 \\
\hline
20 \leq x < 30 & 12 \\
\hline
30 \leq x < 40 & 6 \\
\hline
\end{array}
\][/tex]
Solution:
a) Minimum Number of Dogs with Mass > 24 kg
1. For the mass range [tex]\(20 \leq x < 30\)[/tex]:
- There are 12 dogs in total.
- We need to find the proportion of these dogs that weigh more than 24 kg.
- The range of interest is from 24 kg to 30 kg, which is 6 kg out of the 10 kg range (from 20 kg to 30 kg).
- Therefore, [tex]\( \frac{6}{10} \)[/tex] or 0.6 of these dogs weigh more than 24 kg.
- So, [tex]\( 12 \times 0.6 = 7.2 \)[/tex] dogs.
2. For the mass range [tex]\(30 \leq x < 40\)[/tex]:
- There are 6 dogs in this range.
- All these dogs have a mass greater than 24 kg.
So, minimum number of dogs with mass greater than 24 kg = [tex]\( 7.2 + 6 = 13.2 \)[/tex]
b) Maximum Number of Dogs with Mass > 24 kg
1. For the mass range [tex]\(20 \leq x < 30\)[/tex]:
- All 12 dogs in this range could potentially have a mass greater than 24 kg.
2. For the mass range [tex]\(30 \leq x < 40\)[/tex]:
- Again, all 6 dogs in this range will definitely have a mass greater than 24 kg.
So, the maximum number of dogs with mass greater than 24 kg = [tex]\( 12 + 6 = 18 \)[/tex]
Conclusion:
- The minimum number of dogs that could have a mass of more than 24 kg is [tex]\( \textbf{13.2} \)[/tex].
- The maximum number of dogs that could have a mass of more than 24 kg is [tex]\( \textbf{18} \)[/tex].
