A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.1 years, and standard deviation of 1.6 years. If you randomly purchase one item, what is the probability it will last longer than 10 years? (Give answer to 4 decimal places.)
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.1 years, and standard deviation of 1.6 years. If you randomly purchase one item, what is the probability it will last longer than 10 years? (Give answer to 4 decimal places.)
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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![**Problem Statement:**
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.1 years, and a standard deviation of 1.6 years.
**Question:**
If you randomly purchase one item, what is the probability it will last longer than 10 years? (Give answer to 4 decimal places.)
**Explanation:**
To solve this problem, we need to find the probability that an item will last longer than 10 years. Given that the lifespan is normally distributed, we can use the properties of the normal distribution to find this probability.
1. **Determine the Z-score:**
The Z-score for a value \( x \) in a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) is given by:
\[
Z = \frac{x - \mu}{\sigma}
\]
2. **Use the Z-score to find the probability:**
- Look up the Z-score in the standard normal distribution table, or use a calculator to find the probability.
In this case, we want to determine the probability that an item's lifespan \( x \) is greater than 10 years:
- \( \mu = 11.1 \)
- \( \sigma = 1.6 \)
- \( x = 10 \)
Calculate the Z-score:
\[
Z = \frac{10 - 11.1}{1.6} = \frac{-1.1}{1.6} \approx -0.6875
\]
Look up the Z-score \(-0.6875\) in the Z-table to find the probability that the lifespan is less than 10 years, and subtract it from 1 to find the probability that it is more than 10 years.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbba2c12-57b9-42d4-9dcd-e0986d5f853d%2F9a38c517-db16-4edc-b165-e6b9463468cd%2Fc4sbnt9_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 11.1 years, and a standard deviation of 1.6 years.
**Question:**
If you randomly purchase one item, what is the probability it will last longer than 10 years? (Give answer to 4 decimal places.)
**Explanation:**
To solve this problem, we need to find the probability that an item will last longer than 10 years. Given that the lifespan is normally distributed, we can use the properties of the normal distribution to find this probability.
1. **Determine the Z-score:**
The Z-score for a value \( x \) in a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) is given by:
\[
Z = \frac{x - \mu}{\sigma}
\]
2. **Use the Z-score to find the probability:**
- Look up the Z-score in the standard normal distribution table, or use a calculator to find the probability.
In this case, we want to determine the probability that an item's lifespan \( x \) is greater than 10 years:
- \( \mu = 11.1 \)
- \( \sigma = 1.6 \)
- \( x = 10 \)
Calculate the Z-score:
\[
Z = \frac{10 - 11.1}{1.6} = \frac{-1.1}{1.6} \approx -0.6875
\]
Look up the Z-score \(-0.6875\) in the Z-table to find the probability that the lifespan is less than 10 years, and subtract it from 1 to find the probability that it is more than 10 years.
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