To calculate the arithmetic mean of the given distribution, we'll follow these steps:
1. Find the midpoints of each profit interval: The midpoint of an interval is the average of its lower and upper bounds. For each interval, we calculate:
- [tex]\(0-10\)[/tex]: Midpoint [tex]\((0 + 10) / 2 = 5\)[/tex]
- [tex]\(10-20\)[/tex]: Midpoint [tex]\((10 + 20) / 2 = 15\)[/tex]
- [tex]\(20-30\)[/tex]: Midpoint [tex]\((20 + 30) / 2 = 25\)[/tex]
- [tex]\(30-40\)[/tex]: Midpoint [tex]\((30 + 40) / 2 = 35\)[/tex]
- [tex]\(40-50\)[/tex]: Midpoint [tex]\((40 + 50) / 2 = 45\)[/tex]
- [tex]\(50-60\)[/tex]: Midpoint [tex]\((50 + 60) / 2 = 55\)[/tex]
So, the midpoints are [tex]\([5, 15, 25, 35, 45, 55]\)[/tex].
2. Determine the total number of stalls: Add the number of stalls in each profit interval:
[tex]\[
\text{Total number of stalls} = 126 + 18 + 27 + 20 + 17 + 6 = 214
\][/tex]
3. Calculate the weighted sum of the midpoints: Multiply each midpoint by the corresponding number of stalls, then sum these products:
[tex]\[
\begin{align*}
\text{Weighted sum} &= (5 \times 126) + (15 \times 18) + (25 \times 27) + (35 \times 20) + (45 \times 17) + (55 \times 6) \\
&= 630 + 270 + 675 + 700 + 765 + 330 \\
&= 3370
\end{align*}
\][/tex]
4. Calculate the arithmetic mean: The arithmetic mean is the weighted sum of the midpoints divided by the total number of stalls:
[tex]\[
\text{Arithmetic mean} = \frac{\text{Weighted sum}}{\text{Total number of stalls}} = \frac{3370}{214} \approx 15.748
\][/tex]
Hence, the arithmetic mean of the distribution is approximately [tex]\(15.748\)[/tex].
