To estimate the mean waiting time from a frequency distribution, we need to follow these steps:

1. Identify the midpoints of each class: The midpoint of each class is the average of the lower and upper boundaries of the class.

2. Multiply the midpoint of each class by the frequency: This gives us the "weighted waiting time" for each class.

3. Sum up all the products from step 2: This is the total of all waiting times, weighted by frequency.

4. Divide the total weighted waiting time by the total frequency: This gives the mean waiting time.

Step-by-step:

1. Calculate the midpoints of each class:

• For $0 \text{ to } 6$, midpoint = $\frac{0 + 6}{2} = 3$
• For $7 \text{ to } 13$, midpoint = $\frac{7 + 13}{2} = 10$
• For $14 \text{ to } 20$, midpoint = $\frac{14 + 20}{2} = 17$
• For $21 \text{ to } 27$, midpoint = $\frac{21 + 27}{2} = 24$
• For $28 \text{ to } 34$, midpoint = $\frac{28 + 34}{2} = 31$

2. Assign frequencies for each class:

Since you haven't provided the exact frequencies, I will assign sample frequencies as placeholders:

• $0 \text{ to } 6$: Frequency = $f_1$
• $7 \text{ to } 13$: Frequency = $f_2$
• $14 \text{ to } 20$: Frequency = $f_3$
• $21 \text{ to } 27$: Frequency = $f_4$
• $28 \text{ to } 34$: Frequency = $f_5$

3. Multiply midpoints by frequencies:

• $3 \times f_1$
• $10 \times f_2$
• $17 \times f_3$
• $24 \times f_4$
• $31 \times f_5$

4. Sum up all the products and divide by the total frequency:

$\text{Mean waiting time} = \frac{(3 \times f_1) + (10 \times f_2) + (17 \times f_3) + (24 \times f_4) + (31 \times f_5)}{f_1 + f_2 + f_3 + f_4 + f_5}$

Please provide the frequencies for each class, and I'll complete the calculation for you.

Would you like more clarification on this method, or additional details?

Here are 5 related questions for further exploration:

1. What happens to the mean waiting time if one class has a much higher frequency?
2. How does this approach differ from calculating the median of a frequency distribution?
3. What are the potential limitations of using the midpoint in estimating waiting times?
4. How could Archie improve the precision of his data collection?
5. What other statistical measures could Archie use to strengthen his case?

Tip: Ensure that the frequencies reflect all customers' experiences to avoid sampling bias in your data collection.