Question

A company that manufactures rivets believes the shear strength (in pounds) is modeled by \(N(800,50)\). a) Draw and label the Normal model. b) Would it be safe to use these rivets in a situation requiring a shear strength of 750 pounds? Explain. c) About what percent of these rivets would you expect to fall below 900 pounds? d) Rivets are used in a variety of applications with varying shear strength requirements. What is the maximum shear strength for which you would feel comfortable approving this company's rivets? Explain your reasoning.

Short answer

a) Draw a normal curve centered at 800 with increments of 50. b) Yes, but with caution. c) 97.72%. d) 900 pounds is the practical maximum. Explanation considers distribution.

Step-by-step solution

Understanding the Distribution

The shear strength of the rivets is modeled by a Normal distribution with mean \( \mu = 800 \) pounds and standard deviation \( \sigma = 50 \) pounds. In notation form, \( N(800, 50) \). This description implies rivet strengths are symmetrically distributed around 800 pounds.

Drawing and Labeling the Normal Model

We'll sketch a normal distribution curve centered at 800. The main points to label on the graph are: \( \mu = 800 \), and increments of the standard deviation \( \sigma = 50 \) in both directions: 750 and 850 (one standard deviation away), 700 and 900 (two standard deviations), and 650 and 950 (three standard deviations). This provides a visual representation of the distribution.

Evaluate Safety for 750 Pounds

To determine if a rivet with a shear strength of 750 pounds is safe, we consider its position relative to the distribution. We calculate deviations from the mean: \( 750 = 800 - 50 \), which is exactly \(1\) standard deviation below the mean. Typically, in a normal distribution, about 68% of the data falls within one standard deviation from the mean which includes 750 pounds. This suggests it is somewhat typical, though slightly weaker than average.

Calculate Percentage Below 900 Pounds

To find the percentage of rivets expected to have a strength below 900 pounds, we calculate the z-score: \( z = \frac{900 - 800}{50} = 2 \). Using standard normal distribution tables, a z-score of 2 corresponds to roughly 97.72%. Thus, about 97.72% of rivets are expected to have a shear strength below 900 pounds.

Determine Comfortable Maximum Strength

To determine a maximum safe shear strength, consider the realistic application of rivets and distribution tails. A practical approach is to check up to two standard deviations above the mean (\( 2\sigma = 900 \) pounds), which covers roughly 97.72% of rivets. Beyond this, issues like unusual loads or manufacturing defects could cause failures due to outliers.

Key concepts

Understanding Shear Strength

Shear strength is the ability of a material, like a rivet, to resist forces that can cause it to slide or fail along a plane. In this scenario, shear strength is crucial as it indicates how much stress the rivets can handle before failure. Given the normal distribution model, shear strength is symmetrically distributed around the mean value of 800 pounds, with most data naturally clustering around this central point. Understanding shear strength is vital in applications where mechanical stability and safety are paramount, like in structural buildings or machinery assembly. This helps ensure rivets are strong enough to link materials safely.

The Role of Standard Deviation

Standard deviation is a statistical measure that provides insight into the amount of variation or dispersion in a set of data values. It tells us how much the individual data points, such as rivet strengths, deviate on average from the mean. In the given problem, the standard deviation of 50 pounds describes how spread out the rivet strengths are around the mean of 800 pounds. A smaller standard deviation would imply that the strengths are closely bunched around 800 pounds, indicating more consistent performance. In contrast, a larger standard deviation would suggest a wide range of strengths, reflecting less predictability. Understanding the standard deviation is essential for evaluating how variable the performance of a product like rivets might be, and consequently, planning for safety and reliability.

Interpreting Z-score

The z-score is a statistical measure that describes a value's position relative to the mean of a group of values, measured in terms of standard deviations from the mean. It is used to determine probabilities within a normal distribution. In this exercise, a z-score calculation helps to assess how typical or atypical a shear strength value is, based on its distance from the mean of 800 pounds. For instance, a rivet shear strength of 750 pounds has a z-score of -1, meaning it is one standard deviation below the mean. The z-score can also help determine the percentage of values lying above or below a certain point, as exemplified by calculating that approximately 97.72% of rivets have a strength below 900 pounds. This tool is indispensable for engineers and quality assurance professionals when analyzing how likely specific outcomes are.

Understanding Safety Evaluation

Safety evaluation involves assessing the risks associated with the use of a product under specified conditions. For rivets, this means evaluating if they can reliably hold materials together without failure under normal operating conditions. Safety is often ensured by considering both the mean strength and the variation around it, as indicated by the standard deviation. In this scenario, evaluating whether it is safe to use rivets with a shear strength of 750 pounds involves understanding their position in the distribution and the acceptable risk level in the application context. Typically, products used within two standard deviations of the mean are considered safe, covering around 95% of typical usage outcomes. By closely examining these factors, companies and safety regulators can make informed decisions on the suitability and risk levels of using particular rivets for specific applications. This ensures that the products meet necessary safety standards and performance expectations.