Answered: A mechanic sells a brand of automobile…

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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$### Tire Replacement Guarantee: Practical Application of Normal Distribution A mechanic sells a brand of automobile tire that has a life expectancy that is normally distributed, with a mean life of 36,000 miles and a standard deviation of 2700 miles. He wants to give a guarantee for free replacement of tires that don’t wear well. How should he word his guarantee if he is willing to replace approximately 10% of the tires? #### Instructions: To find the mileage threshold at which 10% of tires wear out prematurely (and are thus eligible for replacement), follow these steps using the standard normal distribution: 1. **Consult the Standard Normal Distribution Table:** - [Click here to view page 1 of the standard normal distribution table.](#) - [Click here to view page 2 of the standard normal distribution table.](#) 2. **Determine the Z-Score for the 10th Percentile:** - Locate the z-score that corresponds to the cumulative probability of 0.10. 3. **Calculate the Respective Mileage:** - Use the mean (36,000 miles) and standard deviation (2700 miles) to convert the z-score into the mileage value. #### Guarantee Statement Tires that wear out by ____ miles will be replaced free of charge. *(Round to the nearest mile as needed.)* --- ### Explanation of the Concepts and Tools #### Standard Normal Distribution Table: The table provides the cumulative probability associated with each z-score in a standard normal distribution (mean = 0, standard deviation = 1). These tables allow us to determine the probability that a sample falls below a particular z-score. #### Calculating Z-Scores: Z-scores indicate how many standard deviations an element is from the mean. They are calculated as follows: \[ Z = \frac{(X - \mu)}{\sigma} \] Where: - $ X $ is the value in the normal distribution. - $ \mu $ is the mean of the distribution. - $ \sigma $ is the standard deviation of the distribution. By finding the z-score corresponding to the 10th percentile and converting it using the formula above, you can determine the mileage at which the mechanic should offer free replacements.$

Transcribed Image Text:### Tire Replacement Guarantee: Practical Application of Normal Distribution A mechanic sells a brand of automobile tire that has a life expectancy that is normally distributed, with a mean life of 36,000 miles and a standard deviation of 2700 miles. He wants to give a guarantee for free replacement of tires that don’t wear well. How should he word his guarantee if he is willing to replace approximately 10% of the tires? #### Instructions: To find the mileage threshold at which 10% of tires wear out prematurely (and are thus eligible for replacement), follow these steps using the standard normal distribution: 1. **Consult the Standard Normal Distribution Table:** - [Click here to view page 1 of the standard normal distribution table.](#) - [Click here to view page 2 of the standard normal distribution table.](#) 2. **Determine the Z-Score for the 10th Percentile:** - Locate the z-score that corresponds to the cumulative probability of 0.10. 3. **Calculate the Respective Mileage:** - Use the mean (36,000 miles) and standard deviation (2700 miles) to convert the z-score into the mileage value. #### Guarantee Statement Tires that wear out by ____ miles will be replaced free of charge. *(Round to the nearest mile as needed.)* --- ### Explanation of the Concepts and Tools #### Standard Normal Distribution Table: The table provides the cumulative probability associated with each z-score in a standard normal distribution (mean = 0, standard deviation = 1). These tables allow us to determine the probability that a sample falls below a particular z-score. #### Calculating Z-Scores: Z-scores indicate how many standard deviations an element is from the mean. They are calculated as follows: \[ Z = \frac{(X - \mu)}{\sigma} \] Where: - $ X $ is the value in the normal distribution. - $ \mu $ is the mean of the distribution. - $ \sigma $ is the standard deviation of the distribution. By finding the z-score corresponding to the 10th percentile and converting it using the formula above, you can determine the mileage at which the mechanic should offer free replacements.

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