A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.9 years, and standard deviation of 0.8 years. If you randomly purchase one item, what is the probability it will last longer than 13 years? Round answer to three decimal places
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.9 years, and standard deviation of 0.8 years. If you randomly purchase one item, what is the probability it will last longer than 13 years? Round answer to three decimal places
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 19PFA
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Question
![**Problem Statement:**
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.9 years and a standard deviation of 0.8 years.
If you randomly purchase one item, what is the probability it will last longer than 13 years?
Round your answer to three decimal places.
---
**Explanation for Educational Context:**
This problem involves the application of statistical concepts, specifically the normal distribution. In this case, you're given a mean lifespan of 13.9 years and a standard deviation of 0.8 years. The task is to find the probability that an item will last longer than 13 years.
To solve this, you would calculate the z-score using the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \( X \) is the value of interest (13 years in this case),
- \( \mu \) is the mean (13.9 years),
- \( \sigma \) is the standard deviation (0.8 years).
After finding the z-score, you can use the standard normal distribution table to find the probability. Since the question asks for the probability of lasting longer than 13 years, you will look for the complement of the probability up to 13 years.
The solution should be rounded to three decimal places as specified.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ebf9e4a-d88a-4068-a98e-7c5e269e8031%2Fd96752d5-5021-42ee-bc8d-7eedfe79b35d%2Fnh4d3o_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.9 years and a standard deviation of 0.8 years.
If you randomly purchase one item, what is the probability it will last longer than 13 years?
Round your answer to three decimal places.
---
**Explanation for Educational Context:**
This problem involves the application of statistical concepts, specifically the normal distribution. In this case, you're given a mean lifespan of 13.9 years and a standard deviation of 0.8 years. The task is to find the probability that an item will last longer than 13 years.
To solve this, you would calculate the z-score using the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \( X \) is the value of interest (13 years in this case),
- \( \mu \) is the mean (13.9 years),
- \( \sigma \) is the standard deviation (0.8 years).
After finding the z-score, you can use the standard normal distribution table to find the probability. Since the question asks for the probability of lasting longer than 13 years, you will look for the complement of the probability up to 13 years.
The solution should be rounded to three decimal places as specified.
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