Problem 193 - Applied Statistics and Probability for Engineers

Question

The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch. (a) What is the probability that the strength of a sample is less than \(40 \mathrm{lb} / \mathrm{in}^{2} ?\) (b) If the specifications require the tensile strength to exceed \(30 \mathrm{lb} / \mathrm{in}^{2},\) what proportion of the samples is scrapped?

Step-by-Step Solution

Answer

(a) Probability: 0.9938, (b) Proportion scrapped: 0.0062

Step 1: Identify the Problem Type

This problem involves a normal distribution with a mean and a standard deviation given. We need to find probabilities for certain conditions related to the tensile strength of paper.

Step 2: Define Given Parameters

The mean (\(\mu\)) is 35 lb/in² and the standard deviation (\(\sigma\)) is 2 lb/in².

Step 3: Convert Measurements to Z-scores for Part (a)

To find the probability for tensile strength less than 40 lb/in², use the Z-score formula:\[ Z = \frac{X - \mu}{\sigma} \]Substitute in the values:\[ Z = \frac{40 - 35}{2} = 2.5 \]

Step 4: Find Probability for Part (a)

Using the Z-table, find the probability corresponding to \( Z = 2.5 \). This gives us \( P(Z < 2.5) \approx 0.9938 \).This means the probability that the strength is less than 40 lb/in² is approximately 0.9938.

Step 5: Convert Measurements to Z-scores for Part (b)

To find the probability for tensile strength not exceeding 30 lb/in², use the Z-score formula:\[ Z = \frac{X - \mu}{\sigma} \]Substitute in the values:\[ Z = \frac{30 - 35}{2} = -2.5 \]

Step 6: Find Probability for Part (b)

Using the Z-table, find the probability corresponding to \( Z = -2.5 \). This gives us \( P(Z < -2.5) \approx 0.0062 \).This means about 0.62%, or 0.0062 of the samples have tensile strength not exceeding 30 lb/in².

Key Concepts

Tensile Strength

When we talk about tensile strength, we're referring to the maximum stress that a material can withstand while being stretched or pulled before breaking. In engineering and materials science, tensile strength is crucial because it tells us how strong the material is.
For paper, tensile strength is typically measured in pounds per square inch (psi). A higher tensile strength means the paper can handle more stress before ripping or tearing. In practical terms, if you're printing on paper or using it in packaging, knowing its tensile strength can help ensure it won't break easily during use.
In the given problem, the tensile strength of the paper is modeled by a normal distribution, which provides a statistical framework to predict how the tensile strength varies across different samples. This helps manufacturers understand and control the quality of the paper produced.

Probability Calculation

To solve problems involving normal distribution, such as finding the probability that a material will meet certain conditions, probability calculations are essential. When you're given a normal distribution with specific characteristics, you can use these to calculate probabilities for specific events.
The problem provided involves calculating the probability that the tensile strength is less than a certain value or that it does not exceed another specified value. Probability gives us a way to express the likelihood of these events occurring.
In probability calculation, we convert physical measurements into Z-scores using a specific formula. By looking up these Z-scores in a Z-table, we find the probability associated with them and can answer questions about how certain materials will perform under different conditions.

Z-score

The Z-score is a way of describing a value's relation to the mean of a group of values. This is crucial in standardizing different data points in a normal distribution. For our tensile strength problem, the Z-score lets us measure how far a sample's tensile strength is from the mean, in terms of standard deviations.
The formula for calculating the Z-score involves subtracting the mean (\(\mu\)) from the observed value (\(X\)) and then dividing by the standard deviation (\(\sigma\)). Mathematically, it's expressed as:\( Z = \frac{X - \mu}{\sigma} \).
In step-by-step problem-solving, once the Z-score is computed, it is used to find corresponding probabilities using a Z-table. This process transforms raw data into a standardized form, making statistical analysis and comparisons simpler.

Statistical Analysis

Statistical analysis involves collecting, reviewing, and interpreting data to uncover patterns or trends. In this case, we are dealing with a problem regarding the tensile strength of paper, modeled by a normal distribution.
By using statistical analysis, we can predict how likely it is for the paper to meet specific tensile strength criteria, which is particularly useful for quality control in manufacturing.

• We analyzed the data using parameters like the mean and standard deviation.
• These parameters help describe the normal distribution and enable the calculation of probabilities for certain outcomes.
• Using statistical tools like the Z-score and Z-table, we can understand the variation in tensile strength and make informed decisions based on these analytical findings.