Let's solve this step by step.
Step 1: Calculate the midpoints of each interval.
To estimate the mean number of visits, we first need the midpoints of each interval:
- For the interval 1 to 7, the midpoint is [tex]\((1 + 7) / 2 = 4\)[/tex].
- For the interval 8 to 14, the midpoint is [tex]\((8 + 14) / 2 = 11\)[/tex].
- For the interval 15 to 21, the midpoint is [tex]\((15 + 21) / 2 = 18\)[/tex].
- For the interval 22 to 28, the midpoint is [tex]\((22 + 28) / 2 = 25\)[/tex].
Step 2: Multiply each midpoint by the corresponding frequency to find the total number of visits.
Using the midpoints calculated:
- For the interval 1 to 7 with frequency 8: [tex]\(4 \times 8 = 32\)[/tex].
- For the interval 8 to 14 with frequency 24: [tex]\(11 \times 24 = 264\)[/tex].
- For the interval 15 to 21 with frequency 12: [tex]\(18 \times 12 = 216\)[/tex].
- For the interval 22 to 28 with frequency 6: [tex]\(25 \times 6 = 150\)[/tex].
Sum all these products to find the total number of visits in August:
[tex]\[ 32 + 264 + 216 + 150 = 662 \][/tex]
Step 3: Calculate the total number of clients.
From the given table, the total number of clients is:
[tex]\[ 8 + 24 + 12 + 6 = 50 \][/tex]
Step 4: Estimate the mean number of visits in August.
To find the mean number of visits in August:
[tex]\[ \text{Mean in August} = \frac{\text{Total number of visits}}{\text{Total number of clients}} = \frac{662}{50} = 13.24 \][/tex]
Step 5: Calculate the mean number of visits in July.
According to the manager, the mean number of visits in August is a 16% increase on the mean number of visits in July. Let [tex]\( x \)[/tex] be the mean number of visits in July. Then, we know:
[tex]\[ 13.24 = x + 0.16x \][/tex]
[tex]\[ 13.24 = 1.16x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{13.24}{1.16} \approx 11.413793103448278 \][/tex]
Therefore, the mean number of client visits in July was approximately 11.41.
So, the step-by-step solution has shown that the mean number of visits in July, before the increase, was approximately [tex]\( 11.41 \)[/tex].
