An elevator has a placard stating that the maximum capacity is 1896 lb—12 passengers. So, 12 adult male passengers can have a mean weight of up to 1896/12=158 pounds. If the elevator is loaded with 12 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 158 lb. (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 25 lb.) Does this elevator appear to be safe?
An elevator has a placard stating that the maximum capacity is 1896 lb—12 passengers. So, 12 adult male passengers can have a mean weight of up to 1896/12=158 pounds. If the elevator is loaded with 12 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 158 lb. (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 25 lb.) Does this elevator appear to be safe?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Question
An elevator has a placard stating that the maximum capacity is
1896
lb—12
passengers. So,
12
adult male passengers can have a
1896/12=158 pounds.
If the elevator is loaded with
12
adult male passengers, find the
158
lb. (Assume that weights of males are
and a standard deviation of
25 lb.)
Does this elevator appear to be safe?
![**Problem Statement:**
An elevator has a placard stating that the maximum capacity is 1896 lb for 12 passengers. Therefore, 12 adult male passengers can have a mean weight of up to \( \frac{1896}{12} = 158 \) pounds. If the elevator is loaded with 12 adult male passengers, calculate the probability that it is overloaded because they have a mean weight greater than 158 lb. (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 25 lb.) Does this elevator appear to be safe?
**Solution Approach:**
To find the probability that the elevator is overloaded, we need to calculate the probability that the mean weight of the 12 passengers exceeds 158 lb.
1. **Given Parameters:**
- Mean weight (\(\mu\)) = 165 lb
- Standard deviation (\(\sigma\)) = 25 lb
- Sample size (\(n\)) = 12
- Maximum allowed mean weight = 158 lb
2. **Calculate the Standard Error (SE) of the Mean:**
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{25}{\sqrt{12}}
\]
3. **Calculate the Z-Score:**
\[
Z = \frac{\text{Observed Mean} - \mu}{SE}
\]
Substitute the values:
\[
Z = \frac{158 - 165}{SE}
\]
4. **Use the Standard Normal Distribution:**
- Look up the Z-score in a standard normal distribution table, or use a calculator to find the probability.
**Conclusion:**
Calculate this probability to determine if the elevator is safe. If the probability is very low, it suggests that the likelihood of the elevator being overloaded is small, indicating that the elevator is safe to use with the given weight assumptions.
The probability the elevator is overloaded is \(\_\_\_\_\). (Round to four decimal places as needed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48e6a7d0-38a2-4f0f-929f-c6198e406298%2Fde01f932-57a0-4084-a8bd-5761ed8802f9%2F81coles_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
An elevator has a placard stating that the maximum capacity is 1896 lb for 12 passengers. Therefore, 12 adult male passengers can have a mean weight of up to \( \frac{1896}{12} = 158 \) pounds. If the elevator is loaded with 12 adult male passengers, calculate the probability that it is overloaded because they have a mean weight greater than 158 lb. (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 25 lb.) Does this elevator appear to be safe?
**Solution Approach:**
To find the probability that the elevator is overloaded, we need to calculate the probability that the mean weight of the 12 passengers exceeds 158 lb.
1. **Given Parameters:**
- Mean weight (\(\mu\)) = 165 lb
- Standard deviation (\(\sigma\)) = 25 lb
- Sample size (\(n\)) = 12
- Maximum allowed mean weight = 158 lb
2. **Calculate the Standard Error (SE) of the Mean:**
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{25}{\sqrt{12}}
\]
3. **Calculate the Z-Score:**
\[
Z = \frac{\text{Observed Mean} - \mu}{SE}
\]
Substitute the values:
\[
Z = \frac{158 - 165}{SE}
\]
4. **Use the Standard Normal Distribution:**
- Look up the Z-score in a standard normal distribution table, or use a calculator to find the probability.
**Conclusion:**
Calculate this probability to determine if the elevator is safe. If the probability is very low, it suggests that the likelihood of the elevator being overloaded is small, indicating that the elevator is safe to use with the given weight assumptions.
The probability the elevator is overloaded is \(\_\_\_\_\). (Round to four decimal places as needed.)
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