To find an estimate of the mean queuing time based on the data provided in the table, we use the midpoints of each time interval as estimates for the queuing times.

1. Step 1: Calculate the midpoints of each time interval:

   • For $0 < t \leq 5$: midpoint = $\frac{0 + 5}{2} = 2.5$
   • For $5 < t \leq 10$: midpoint = $\frac{5 + 10}{2} = 7.5$
   • For $10 < t \leq 15$: midpoint = $\frac{10 + 15}{2} = 12.5$
   • For $15 < t \leq 20$: midpoint = $\frac{15 + 20}{2} = 17.5$

2. Step 2: Multiply the midpoints by their respective frequencies to get the total time for each group:

   • $2.5 \times 14 = 35$
   • $7.5 \times 23 = 172.5$
   • $12.5 \times 8 = 100$
   • $17.5 \times 5 = 87.5$

3. Step 3: Find the total time for all groups: $35 + 172.5 + 100 + 87.5 = 395$

4. Step 4: Find the total frequency, which is the total number of customers: $14 + 23 + 8 + 5 = 50$

5. Step 5: Estimate the mean queuing time by dividing the total time by the total frequency: $\text{Mean} = \frac{395}{50} = 7.9 \, \text{minutes}$

Thus, the estimated mean queuing time is 7.9 minutes.

Would you like further details on this? Here are 5 follow-up questions you can explore:

1. How would the estimate change if the time intervals were narrower?
2. Why do we use midpoints in estimating mean times for grouped data?
3. How can we visualize this data using a histogram?
4. What assumptions are made in estimating the mean from grouped data?
5. How would we estimate the median queuing time from this data?

Tip: Always check that the sum of the frequencies matches the total sample size when working with grouped data!