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A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 207.2-cm and a standard deviation of 1.5-cm. For shipment, 47 steel rods are bundled together. Round all answers to four decimal places if necessary. a. What is the distribution of X? X - N( b. What is the distribution of I? I - N c. For a single randomly selected steel rod, find the probability that the length is between 207-cm and 207.1-cm. d. For a bundled of 47 rods, find the probability that the average length is between 207-cm and 207.1- cm. e. For part d), is the assumption of normal necessary? No Yes 4

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 207.2-cm and a standard deviation of 1.5-cm. For shipment, 47 steel rods are bundled together. Round all answers to four decimal places if necessary. a. What is the distribution of X? X - N( b. What is the distribution of I? I - N c. For a single randomly selected steel rod, find the probability that the length is between 207-cm and 207.1-cm. d. For a bundled of 47 rods, find the probability that the average length is between 207-cm and 207.1- cm. e. For part d), is the assumption of normal necessary? No Yes 4

MATLAB: An Introduction with Applications
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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**Steel Rod Length Distribution and Probability Calculation** A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 207.2 cm and a standard deviation of 2.7 cm. For shipment, 47 steel rods are bundled together. *Round all answers to four decimal places if necessary.* ### Problem Statement 1. **Distribution of Individual Steel Rod Lengths:** - What is the distribution of \( X \)? - Answer: \( X \sim N(\mu, \sigma) \) - Substituting the given values: \( X \sim N(207.2, 2.7) \) 2. **Distribution of Mean Length of 47 Steel Rods:** - What is the distribution of \( \bar{X} \)? - Answer: \( \bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right) \) - Substituting the given values: \( \bar{X} \sim N\left(207.2, \frac{2.7}{\sqrt{47}}\right) \) 3. **Probability Calculations:** - **For a single randomly selected steel rod, find the probability that the length is between 207 cm and 207.1 cm.** - Calculation: \( P(207 \leq X \leq 207.1) \) - **For a bundle of 47 rods, find the probability that the average length is between 207 cm and 207.1 cm.** - Calculation: \( P(207 \leq \bar{X} \leq 207.1) \) 4. **Assumption Check:** - **For part d, is the assumption of normal necessary? (YES/NO)** - Answer: YES ### Important Concepts: - **Normal Distribution:** - Represents data that clusters around a mean or average. - Symmetrical, bell-shaped distribution. - **Mean (μ):** - The average of all the data points. - **Standard Deviation (σ):** - Measures the dispersion or spread of the data points from the mean. - **Central Limit Theorem:** - For a large enough sample size, the sampling distribution of the sample mean will be normally distributed regardless of the shape of
Transcribed Image Text:**Steel Rod Length Distribution and Probability Calculation** A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 207.2 cm and a standard deviation of 2.7 cm. For shipment, 47 steel rods are bundled together. *Round all answers to four decimal places if necessary.* ### Problem Statement 1. **Distribution of Individual Steel Rod Lengths:** - What is the distribution of \( X \)? - Answer: \( X \sim N(\mu, \sigma) \) - Substituting the given values: \( X \sim N(207.2, 2.7) \) 2. **Distribution of Mean Length of 47 Steel Rods:** - What is the distribution of \( \bar{X} \)? - Answer: \( \bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right) \) - Substituting the given values: \( \bar{X} \sim N\left(207.2, \frac{2.7}{\sqrt{47}}\right) \) 3. **Probability Calculations:** - **For a single randomly selected steel rod, find the probability that the length is between 207 cm and 207.1 cm.** - Calculation: \( P(207 \leq X \leq 207.1) \) - **For a bundle of 47 rods, find the probability that the average length is between 207 cm and 207.1 cm.** - Calculation: \( P(207 \leq \bar{X} \leq 207.1) \) 4. **Assumption Check:** - **For part d, is the assumption of normal necessary? (YES/NO)** - Answer: YES ### Important Concepts: - **Normal Distribution:** - Represents data that clusters around a mean or average. - Symmetrical, bell-shaped distribution. - **Mean (μ):** - The average of all the data points. - **Standard Deviation (σ):** - Measures the dispersion or spread of the data points from the mean. - **Central Limit Theorem:** - For a large enough sample size, the sampling distribution of the sample mean will be normally distributed regardless of the shape of
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