Problem 28 - Basic Statistics for Business and Economics

Question

The monthly sales of mufflers in the Richmond, Virginia, area follow the normal distribution with a mean of 1,200 and a standard deviation of \(225 .\) The manufacturer would like to establish inventory levels such that there is only a 5 percent chance of running out of stock. Where should the manufacturer set the inventory levels?

Step-by-Step Solution

Answer

Set the inventory level to approximately 1570 mufflers.

Step 1: Understand the Problem

We are given that the monthly sales of mufflers follow a normal distribution with mean \( \mu = 1200\) and standard deviation \( \sigma = 225 \). We want to determine the inventory level such that there is only a 5% chance of running out of stock. This corresponds to finding the 95th percentile of the normal distribution.

Step 2: Find the Z-Score

We need to find the Z-score that corresponds to the 95th percentile of a standard normal distribution. From Z-tables or using statistical software, we know that the Z-score for the 95th percentile is approximately \(1.645\).

Step 3: Calculate Inventory Level

Using the Z-score formula, \[ X = \mu + Z \times \sigma \]Substitute the values: \[ X = 1200 + 1.645 \times 225 \]

Step 4: Compute the Result

Calculate \( X \):\[ X = 1200 + 370.125 = 1570.125 \]Therefore, the manufacturer should set the inventory level to approximately 1570 units.

Key Concepts

Inventory Management

Inventory management is a critical aspect of running a successful business, especially for companies dealing with tangible goods. The goal is to have just enough stock to meet customer demand without overstocking, which ties up valuable capital. Setting inventory levels strategically involves predicting demand to avoid stockouts and overstock.

• **Understanding Demand Patterns**: Businesses can use historical sales data to identify trends and predict future demand. By knowing that muffler sales have a mean of 1,200 units and a standard deviation of 225, the manufacturer can anticipate variations in sales volume.
• **Risk Management**: Companies aim to minimize the risk of running out of stock. For instance, by setting inventory at the 95th percentile, as in our exercise, the manufacturer ensures they'll likely meet customer demand 95% of the time.
• **Optimization**: Efficient inventory management not only saves money but also reduces waste and storage costs. Using tools like Z-scores in conjunction with sales data can help in making data-driven decisions.

Effective inventory management requires a careful balance between cost and service level, ensuring customer satisfaction while maintaining profitability.

Percentiles

Percentiles are used to understand and interpret data points within a distribution. A percentile indicates the value below which a given percentage of observations fall in a dataset.

• **Application in Business**: Percentiles are valuable in inventory management to set stock levels. For example, the 95th percentile ensures coverage of up to 95% of likely sales scenarios, allowing businesses to prepare for high-demand periods without excess inventory.
• **Calculating the 95th Percentile**: In the normal distribution from the exercise, the 95th percentile can be found using a Z-score, derived from tables or statistical software.

The percentile provides a practical approach for business strategists to handle variations in demand and make informed decisions.

Z-Score Calculation

A Z-score is a statistical measure that quantifies the distance a data point is from the mean, in terms of standard deviations. It is a crucial tool in statistics and data analysis.

• **Standard Normal Distribution**: A Z-score transforms data into a standard normal distribution with a mean of 0 and a standard deviation of 1. This allows for the comparison of data from different normal distributions.
• **Finding Z-scores**: Z-scores are found using the standard normal distribution tables or software, correlating percentiles to their respective Z-scores.

For our example, the process involves finding the Z-score for the 95th percentile, which is approximately 1.645. This value is then used to calculate the inventory level by plugging it into the formula:
\[ X = \mu + Z \times \sigma \] To find that the inventory level, in this case, should be set to approximately 1570 units.