Questions: Approximating the mean of a data set given a histogram Archie is fed up with waiting in line at his local post office and decides to take action. Over the course of the next few months, Archie records the waiting times for each of a random selection of 28 post office visits made by him and other customers. These waiting times (in minutes) are summarized in the following histogram. Based on the histogram, estimate the mean waiting time (in minutes) for Archie's sample. Carry your intermediate computations to at least four decimal places, and round your answer to at least one decimal place.

Solution Steps

To estimate the mean waiting time, we first need to identify the midpoints of each interval in the histogram. The intervals and their midpoints are:

• 0-6 minutes: midpoint = (0 + 6) / 2 = 3
• 6-12 minutes: midpoint = (6 + 12) / 2 = 9
• 12-18 minutes: midpoint = (12 + 18) / 2 = 15
• 18-24 minutes: midpoint = (18 + 24) / 2 = 21
• 24-30 minutes: midpoint = (24 + 30) / 2 = 27

Next, we multiply each midpoint by the frequency of its corresponding interval:

• 3 minutes * 4 = 12
• 9 minutes * 10 = 90
• 15 minutes * 5 = 75
• 21 minutes * 6 = 126
• 27 minutes * 3 = 81

We then sum the products and the frequencies:

• Sum of products: 12 + 90 + 75 + 126 + 81 = 384
• Sum of frequencies: 4 + 10 + 5 + 6 + 3 = 28

Finally, we calculate the mean waiting time by dividing the sum of the products by the sum of the frequencies:

• Mean waiting time = 384 / 28 ≈ 13.7143

Final Answer