Problem 185 - Applied Statistics and Probability for Engineers

Question

Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that \(2 \%\) and \(1 \%\), respectively, of the sample shipped in small and large packages, respectively, break during transit. If \(60 \%\) of the samples are shipped in large packages and \(40 \%\) are shipped in small packages, what proportion of samples break during shipment?

Step-by-Step Solution

Answer

1.4% of the samples break during shipment.

Step 1: Understand the Problem

We need to determine the overall proportion of samples that break during shipment. We have two types of packages: small and large. For small packages, 2% break, and for large packages, 1% break. Also, 40% of the samples are in small packages, and 60% are in large packages.

Step 2: Calculate Breakage Rate for Each Package Type

For small packages, the breakage rate is 2% of 40%, and for large packages, it is 1% of 60%.

Step 3: Compute Breakage Proportion for Small Packages

The breakage proportion for small packages is calculated as follows: \[0.40 \times 0.02 = 0.008\]This means that 0.8% of all samples are small packages that break.

Step 4: Compute Breakage Proportion for Large Packages

The breakage proportion for large packages is calculated as follows:\[0.60 \times 0.01 = 0.006\]This means that 0.6% of all samples are large packages that break.

Step 5: Add the Proportions to Find Total Breakage Rate

Add the breakage rates from small and large packages to find the total breakage rate:\[0.008 + 0.006 = 0.014\]Thus, 1.4% of all samples break during shipment.

Key Concepts

Breakage Rate

The breakage rate represents the percentage of items in a shipment that are damaged or broken. This is crucial for businesses, especially those dealing with fragile products like laboratory glassware. When discussing breakage rate, this concept refers to the probability that an individual package will sustain damage during transit.
The breakage rate helps companies determine potential loss and informs decisions on packaging choices.
In our problem, the breakage rate differs for two types of packages: small packages have a breakage rate of 2%, and large packages have a breakage rate of 1%. Understanding these rates can assist in figuring out the total percentage of broken items from the entire shipment.

Package Types

Package types refer to the different forms or designs of packaging used to ship items, based on their size, weight, and fragility. In our exercise, there are two package types:

• Small, light packages
• Heavy, large packages

The type of package can directly influence the breakage rate due to factors like the level of protection it offers and its susceptibility to damage. For example, smaller packages might seem more at risk if not properly cushioned. Moreover, larger packages may naturally offer more protection due to their size and build.
Understanding the characteristics of each package type provides a better insight into predicting and managing breakage rates.

Shipment Proportion

Shipment proportion is the division of different package types within the overall shipment. It indicates how many of each package type are included in the total shipment.

• In our example, 60% of the samples are shipped in large packages.
• Meanwhile, 40% are shipped in small packages.

Shipment proportion helps us calculate the total breakage rate for the entire shipment.
If most of the shipment consists of a package type with a higher breakage rate, the overall loss could be significant. However, a balanced proportion of package types with lower breakage rates can minimize total shipment damage. The interplay between shipment proportion and breakage rate determines the reliability of shipping strategies.

Mathematical Calculation

Mathematical calculation involves the use of arithmetic to find specific results from given data. In this exercise, calculation is vital to find the total breakage rate.
Two calculations were central: the breakage proportions for small and large packages. These were done through simple multiplication.

• The breakage proportion for small packages is 2% of 40%, or \[0.40 \times 0.02 = 0.008\]. This equates to 0.8% overall.
• For large packages, the breakage proportion is 1% of 60%, or \[0.60 \times 0.01 = 0.006\]. Thus, 0.6% overall.

Adding these results—0.8% from small packages and 0.6% from large packages—gives a total breakage rate of 1.4%.
The strength of these mathematical calculations lies in their ability to provide precise insights into business operations and logistics management, showing the significance of mathematical knowledge in practical scenarios.