Problem 52 - Basic Statistics for Business and Economics with Student

Question

New Process, Inc., a large mail-order supplier of women's fashions, advertises same-day service on every order. Recently the movement of orders has not gone as planned, and there were a large number of complaints. Bud Owens, director of customer service, has completely redone the method of order handling. The goal is to have fewer than five unfilled orders on hand at the end of 95 percent of the working days. Frequent checks of the unfilled orders at the end of the day reveal that the distribution of the unfilled orders follows a Poisson distribution with a mean of two orders. a. Has New Process, Inc., lived up to its internal goal? Cite evidence b. Draw a histogram representing the Poisson probability distribution of unfilled orders.

Step-by-Step Solution

Answer

New Process, Inc. did not meet its goal as the probability is 94.73%.

Step 1: Identify the Problem

We need to determine if fewer than five unfilled orders occur on less than or equal to 95% of working days. The distribution is Poisson with a mean (\(\lambda\)) of 2 orders.

Step 2: Understand the Poisson Distribution

The Poisson distribution is used to model the number of events happening within a fixed interval of time. The probability of observing k unfilled orders is given by the formula: \[P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!}\] where \(\lambda = 2\).

Step 3: Calculate Cumulative Probability for k < 5

We calculate the cumulative probability of having fewer than 5 unfilled orders using the Poisson distribution formula:\[P(X < 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)\]Compute for each:\[P(X=0) = \frac{e^{-2} \cdot 2^{0}}{0!}\]\[P(X=1) = \frac{e^{-2} \cdot 2^{1}}{1!}\]\[P(X=2) = \frac{e^{-2} \cdot 2^{2}}{2!}\]\[P(X=3) = \frac{e^{-2} \cdot 2^{3}}{3!}\]\[P(X=4) = \frac{e^{-2} \cdot 2^{4}}{4!}\]

Step 4: Compute Probabilities

Evaluate the probabilities:\[P(X=0) = \frac{e^{-2}}{1} \approx 0.1353\]\[P(X=1) = \frac{2e^{-2}}{1} \approx 0.2707\]\[P(X=2) = \frac{4e^{-2}}{2} \approx 0.2707\]\[P(X=3) = \frac{8e^{-2}}{6} \approx 0.1804\]\[P(X=4) = \frac{16e^{-2}}{24} \approx 0.0902\]

Step 5: Total Cumulative Probability

Sum the individual probabilities for fewer than 5 unfilled orders to find:\[P(X < 5) = 0.1353 + 0.2707 + 0.2707 + 0.1804 + 0.0902 = 0.9473\]

Step 6: Evaluate Company Goal

The probability of having fewer than 5 unfilled orders is approximately 94.73%, which does not meet the internal company goal of 95%. Therefore, New Process, Inc., has not lived up to its internal goal.

Step 7: Draw Histogram

To create a histogram, plot a bar graph where the x-axis represents the possible number of unfilled orders (0 through 4 and possibly 5 for completeness) and the y-axis represents their respective probabilities. Use the calculated probabilities to plot the height of each bar.

Key Concepts

Cumulative Probability

Cumulative probability is a crucial concept when analyzing distributions, such as the Poisson distribution in this case.
It specifically deals with the probability that a random variable takes on a value less than or equal to a certain point.
This is especially vital in determining thresholds and assessing outcomes over a range of values. In the context of the problem faced by New Process, Inc., cumulative probability helps us to determine if they meet their business goal.
By calculating the cumulative probability of having fewer than five unfilled orders, they can assess their performance against the target of 95% working days.
Essentially, we sum up the probabilities of having 0, 1, 2, 3, and 4 unfilled orders to get the cumulative probability. This value offers insight into the overall effectiveness of the company's new order handling method.
It's a measure that aggregates the likelihood of different order counts and helps in making strategic decisions based on statistical analysis.

Probability Distribution

A probability distribution is a statistical function that describes the likelihood of different outcomes in an experiment.
In this exercise, we focus on the Poisson probability distribution, which is particularly useful for modeling the number of events in a fixed interval of time or space.The Poisson distribution is defined by the parameter \(\lambda\), which represents the average rate of occurrence.
This distribution assumes that events happen independently, and the probability of a given number of events occurring is determined by:\[ P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!} \]Here, \(\lambda = 2\) for New Process, Inc., representing the mean number of unfilled orders.
Each value of \(k\) reflects possible counts of unfilled orders, with probabilities diminishing as \(k\) increases.Visualizing this distribution with a histogram helps to see the spread and probabilities of various outcomes visually.
A well-defined probability distribution like the Poisson provides a solid foundation for analyzing data and predicting future outcomes.

Statistical Analysis

Statistical analysis involves collecting, exploring, and drawing insights from data.
For New Process, Inc., this technique is essential to understand and improve their order handling process. Through statistical analysis, companies can view key patterns and probabilities to ensure process efficiency.
This involves using mathematical formulas and models, like the Poisson distribution, to represent real-world processes. In the exercise, the analysis starts by collating data on unfilled orders.
By applying the Poisson formula, probabilities for different order levels are discovered, enabling a deeper analysis.

• Identifying potential gaps or inefficiencies through data comparison.
• Drawing insights into how often certain outcomes occur.
• Evaluating performance against goals, like the 95% threshold, to highlight areas for improvement.

Ultimately, statistical analysis aids organizations in making informed decisions, mitigating risks, and optimizing processes for better customer satisfaction. This empowers New Process, Inc. to take actionable steps toward enhancing their operational strategy based on comprehensive, data-driven insights.