Normal with mean 22.8 inches and standard devia- tion 1.1 inches. (a) A male soldier whose head circumference is 23.9 inches would be at what percentile? Show your method clearly. (b) The army's helmet supplier regularly stocks hel- mets that fit male soldiers with head circumfer- ences between 20 and 26 inches. Anyone with a head circumference outside that interval requires a customized helmet order. What percent of male soldiers require custom helmets? (c) Find the interquartile range for the distribution of head circumference among male soldiers.
Normal with mean 22.8 inches and standard devia- tion 1.1 inches. (a) A male soldier whose head circumference is 23.9 inches would be at what percentile? Show your method clearly. (b) The army's helmet supplier regularly stocks hel- mets that fit male soldiers with head circumfer- ences between 20 and 26 inches. Anyone with a head circumference outside that interval requires a customized helmet order. What percent of male soldiers require custom helmets? (c) Find the interquartile range for the distribution of head circumference among male soldiers.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![**T2.12** The army reports that the distribution of head circumference among male soldiers is approximately
[No graphs or diagrams are present in the image.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b2ad7e-71be-4993-92e0-fe2030354ee3%2Fe2587cf6-b60a-4bf4-90b4-7e14b97677b5%2Fm51ghw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**T2.12** The army reports that the distribution of head circumference among male soldiers is approximately
[No graphs or diagrams are present in the image.]
![**Understanding Helmet Fit: A Statistical Analysis for Male Soldiers**
In this educational module, we explore the distribution of head circumferences among male soldiers, assuming a normal distribution with a mean of 22.8 inches and a standard deviation of 1.1 inches. This analysis helps in understanding helmet fitting requirements.
---
**(a) Percentile Calculation:**
To find the percentile at which a male soldier with a head circumference of 23.9 inches falls, you need to calculate the Z-score and then use a standard normal distribution table or calculator.
1. **Calculate the Z-score:**
\[
Z = \frac{X - \mu}{\sigma} = \frac{23.9 - 22.8}{1.1} \approx 1.00
\]
2. **Determine the Percentile:**
Using a Z-table or calculator, find the percentile for \(Z = 1.00\).
---
**(b) Helmet Stock Requirements:**
The army stocks helmets for head circumferences between 20 and 26 inches. Soldiers with head circumferences outside this range require custom helmets. To find the percentage of soldiers needing customization, calculate:
1. **Z-scores for boundaries:**
- \( Z_{20} = \frac{20 - 22.8}{1.1} \approx -2.55 \)
- \( Z_{26} = \frac{26 - 22.8}{1.1} \approx 2.91 \)
2. **Calculate the cumulative probabilities for these Z-scores** and find the proportion of soldiers whose head circumferences fall outside these limits.
---
**(c) Interquartile Range (IQR):**
To find the interquartile range of head circumferences:
1. **Determine the Z-scores for the 25th and 75th percentiles:**
- \( Q1: Z \approx -0.675 \)
- \( Q3: Z \approx 0.675 \)
2. **Convert these Z-values to head circumferences:**
- \( Q1 = \mu + Z \times \sigma = 22.8 + (-0.675 \times 1.1) \)
- \( Q3 = \mu + Z \times \sigma = 22.8 + (0.675 \times 1.1) \)
3. **](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b2ad7e-71be-4993-92e0-fe2030354ee3%2Fe2587cf6-b60a-4bf4-90b4-7e14b97677b5%2Fcrube33_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Helmet Fit: A Statistical Analysis for Male Soldiers**
In this educational module, we explore the distribution of head circumferences among male soldiers, assuming a normal distribution with a mean of 22.8 inches and a standard deviation of 1.1 inches. This analysis helps in understanding helmet fitting requirements.
---
**(a) Percentile Calculation:**
To find the percentile at which a male soldier with a head circumference of 23.9 inches falls, you need to calculate the Z-score and then use a standard normal distribution table or calculator.
1. **Calculate the Z-score:**
\[
Z = \frac{X - \mu}{\sigma} = \frac{23.9 - 22.8}{1.1} \approx 1.00
\]
2. **Determine the Percentile:**
Using a Z-table or calculator, find the percentile for \(Z = 1.00\).
---
**(b) Helmet Stock Requirements:**
The army stocks helmets for head circumferences between 20 and 26 inches. Soldiers with head circumferences outside this range require custom helmets. To find the percentage of soldiers needing customization, calculate:
1. **Z-scores for boundaries:**
- \( Z_{20} = \frac{20 - 22.8}{1.1} \approx -2.55 \)
- \( Z_{26} = \frac{26 - 22.8}{1.1} \approx 2.91 \)
2. **Calculate the cumulative probabilities for these Z-scores** and find the proportion of soldiers whose head circumferences fall outside these limits.
---
**(c) Interquartile Range (IQR):**
To find the interquartile range of head circumferences:
1. **Determine the Z-scores for the 25th and 75th percentiles:**
- \( Q1: Z \approx -0.675 \)
- \( Q3: Z \approx 0.675 \)
2. **Convert these Z-values to head circumferences:**
- \( Q1 = \mu + Z \times \sigma = 22.8 + (-0.675 \times 1.1) \)
- \( Q3 = \mu + Z \times \sigma = 22.8 + (0.675 \times 1.1) \)
3. **
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