Problem 12 - Basic Statistics for Business and Economics

Question

Ecommerce.com, a large Internet retailer, is studying the lead time (elapsed time between when an order is placed and when it is filled) for a sample of recent orders. The lead times are reported in days. $$ \begin{array}{|cc|} \hline \text { Lead Time (days) } & \text { Frequency } \\ \hline 0 \text { up to } 5 & 6 \\ 5 \text { up to } 10 & 7 \\ 10 \text { up to } 15 & 12 \\ 15 \text { up to } 20 & 8 \\ 20 \text { up to } 25 & \frac{7}{40} \\ \text { Total } & 40 \\ \hline \end{array} $$ a. How many orders were studied? b. What is the midpoint of the first class? c. What are the coordinates of the first class for a frequency polygon? d. Draw a histogram. e. Draw a frequency polygon. f. Interpret the lead times using the two charts.

Step-by-Step Solution

Answer

a. 40 orders. b. Midpoint is 2.5. c. Coordinates are (2.5, 6). d & e. Draw histograms and polygons based on provided frequencies. f. Most lead times are under 20 days, peaking at 10-15 days.

Step 1: Determine Total Orders

Add all the frequencies to find the total number of orders that were studied. The frequencies given are: 6, 7, 12, 8, and (7/40)*40=7.\[ 6 + 7 + 12 + 8 + 7 = 40 \] Therefore, a total of 40 orders were studied.

Step 2: Calculate the Midpoint of the First Class

To find the midpoint of a class, average the lower and upper bounds of the class interval. For the first class having bounds 0 and 5:\[ \text{Midpoint} = \frac{0 + 5}{2} = 2.5 \]Thus, the midpoint of the first class is 2.5 days.

Step 3: Define Frequency Polygon Coordinates for the First Class

A frequency polygon uses midpoints as the x-values and frequencies as the y-values. For the first class: Midpoint = 2.5 (from step 2) Frequency = 6 The coordinates are (2.5, 6).

Step 4: Draw the Histogram

To draw a histogram: 1. On the x-axis, mark the lead time intervals: 0-5, 5-10, 10-15, 15-20, 20-25. 2. On the y-axis, mark the frequencies: 6, 7, 12, 8, 7. 3. Draw bars for each class interval with height corresponding to the frequency. - 0-5 interval: bar height is 6. - 5-10 interval: bar height is 7. - 10-15 interval: bar height is 12. - 15-20 interval: bar height is 8. - 20-25 interval: bar height is 7.

Step 5: Draw the Frequency Polygon

To draw a frequency polygon: 1. Plot points using the midpoints and frequencies: - (2.5, 6), (7.5, 7), (12.5, 12), (17.5, 8), (22.5, 7). 2. Connect these points with straight lines. 3. Optionally, extend the line to the x-axis before the first point (at midpoint -2.5) and after the last point (at midpoint 27.5) with a frequency of 0.

Step 6: Interpret the Charts

From the histogram and frequency polygon, observe: 1. Most orders are filled between 10 to 15 days, indicated by the highest frequency. 2. The lead times tend to be less than 20 days for most of the orders. 3. Few orders have lead times beyond 20 days, as shown by the lower frequencies for higher lead time intervals.

Key Concepts

Lead Time Analysis

Lead time analysis involves examining the elapsed time between placing an order and fulfilling it. In this exercise, Ecommerce.com records the lead times of 40 sample orders, noting how long it takes for each to be completed. This analysis helps identify patterns or potential delays in the order fulfillment process. Understanding lead time is crucial for improving operational efficiency and customer satisfaction. By analyzing the frequency and distribution of lead times, the company can identify areas that may need process improvements. This analysis is often performed with statistical methods such as constructing frequency distributions to summarize and interpret data effectively. The goal is to see trends, ensure prompt service, and maintain a competitive edge in the market.

Histogram

A histogram is a graphical representation of data distribution which allows us to understand the frequency of different outcomes in a dataset. For Ecommerce.com, the histogram depicts lead time frequencies in varying day intervals. Each bar’s height in a histogram indicates the frequency of data points within a particular range:

• 0-5 days with a frequency of 6
• 5-10 days with a frequency of 7
• 10-15 days with a frequency of 12
• 15-20 days with a frequency of 8
• 20-25 days with a frequency of 7

These intervals show how lead times are spread out over different periods. With this visual, the density of orders in each period becomes apparent, aiding in quickly spotting trends or anomalies in lead times, often revealing insights into process efficiencies or customer service levels.

Frequency Polygon

A frequency polygon is similar to a histogram but uses points connected by straight lines to display data distribution. The x-coordinates are class midpoints, and y-coordinates are frequencies, helping to visualize trends across continuous intervals. For Ecommerce.com's lead times, the frequency polygon connects points like (2.5, 6) for 0-5 days and (7.5, 7) for 5-10 days, forming a line graph. By connecting these midpoints:

• 0-5 days midpoint 2.5, frequency 6
• 5-10 days midpoint 7.5, frequency 7
• 10-15 days midpoint 12.5, frequency 12
• 15-20 days midpoint 17.5, frequency 8
• 20-25 days midpoint 22.5, frequency 7

It creates a smooth line that reveals the shape of data distribution. This representation is particularly useful for identifying shifts or peaks over time, allowing Ecommerce.com to see clearer trends and fluctuations in lead times.

Data Interpretation

Interpreting the data presented by histograms and frequency polygons allows us to derive valuable insights. In the case of Ecommerce.com, using these charts reveals critical information about their lead times. The histogram shows that most orders are fulfilled within 10 to 15 days, indicating this is when the company operates most efficiently. Both charts highlight that lead times generally do not exceed 20 days, suggesting that the company can manage its logistics effectively most of the time. However, occasionally longer lead times are seen past 20 days, which prompts further investigation into process optimization. These insights help businesses like Ecommerce.com to assess their performance and make data-driven decisions aimed at process improvements and customer satisfaction.