### Understanding Normal Distribution in ManufacturingWhen assessing the quality and dimensions of manufactured items, it is important to understand the concept of normal distribution. We know that a manufacturer has determined that the lengths of their items are normally distributed. This means that the lengths follow a bell-shaped curve when plotted, with most items clustering around a central mean value and progressively fewer items occurring as the length deviates from this mean.Here’s an important scenario:#### ScenarioA manufacturer knows that their items have a normally distributed length, with a **mean** of 9.8 inches, and a **standard deviation** of 0.8 inches.**Question:**If one item is chosen at random, what is the probability that it is less than 11.9 inches long?#### Concept ExplanationTo determine this probability, we can utilize the properties of the normal distribution. The **mean (µ)** of the distribution is 9.8 inches, and the **standard deviation (σ)** is 0.8 inches. The probability question requires us to find the likelihood that a randomly chosen item will have a length less than 11.9 inches.#### Steps to Solution1. **Convert the value to a Z-score**: The Z-score represents the number of standard deviations a particular value is from the mean. The formula for the Z-score is: \[ Z = \frac{(X - \mu)}{\sigma} \] Where: - \(X\) is the value (11.9 inches) - \(\mu\) is the mean (9.8 inches) - \(\sigma\) is the standard deviation (0.8 inches)2. **Calculate the Z-score for 11.9 inches**: \[ Z = \frac{(11.9 - 9.8)}{0.8} = \frac{2.1}{0.8} = 2.625 \]3. **Use Z-tables to find the probability**: Once we have the Z-score, we use standard normal distribution tables (Z-tables) or a calculator to find the probability that Z is less than 2.625.4. **Interpret the result**: The Z-score of 2.625 corresponds to a cumulative probability of approximately 0.9956 (or 99.56%).Therefore, the probability that a randomly chosen item is less than

Here the length of item follows Normal distribution with mean=9.8 and standard deviation=0.8, ie…

A manufacturer knows that their items have a normally distributed length, with a mean of 9.8 inches, and standard deviation of 0.8 inches. If one item is chosen at random, what is the probability that it is less than 11.9 inches long?

A manufacturer knows that their items have a normally distributed length, with a mean of 9.8 inches, and standard deviation of 0.8 inches. If one item is chosen at random, what is the probability that it is less than 11.9 inches long?

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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