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Chapter 6: Problem 24

Use these parameters (based on Data Set 1 "Body Data" in Appendix \(B\) ): Men's heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in. Women's heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in. The Gulfstream 100 is an executive jet that seats six, and it has a doorway height of 51.6 in. a. What percentage of adult men can fit through the door without bending? b. Does the door design with a height of 51.6 in. appear to be adequate? Why didn't the engineers design a larger door? c. What doorway height would allow \(40 \%\) of men to fit without bending?

Short Answer

Expert verified
a. 0% of adult men can fit without bending. b. The door's height is inadequate for men; constraints likely influenced size. c. A doorway height of 67.9 in. allows 40% of men to fit without bending.

Step by step solution

01

Calculate the Z-score for Men's Height

The Z-score formula is given by: \[ Z = \frac{X - \mu}{\sigma} \]Where, \( X = 51.6 \text{ in (doorway height)} \)\( \mu = 68.6 \text{ in (mean height of men)} \)\( \sigma = 2.8 \text{ in (standard deviation of men's height)} \)Substitute the values into the formula:\[ Z = \frac{51.6 - 68.6}{2.8} = \frac{-17}{2.8} \approx -6.07 \]
02

Determine the Percentage of Men Who Can Fit Without Bending

Using the Z-table or standard normal distribution table, a Z-score of -6.07 is extremely low, meaning the probability of a Z-score being less than -6.07 is nearly 0%. This indicates that essentially 0% of men can fit through the door without bending.
03

Analyze the Door's Adequacy

Given that almost 0% of adult men can fit through the 51.6-inch doorway without bending, this doorway height does not seem adequate for adult men. Engineers likely didn't design a taller door due to constraints such as aerodynamics, construction costs, or weight limitations for the jet.
04

Calculate the Doorway Height for 40% of Men to Fit Without Bending

To find the doorway height that allows 40% of men to fit without bending, we find the Z-score for the 40th percentile. Using the Z-table, the Z-score corresponding to the 40th percentile is approximately -0.25.Using the Z-score formula backwards:\[ X = Z \cdot \sigma + \mu \]Substitute \( Z = -0.25 \), \( \sigma = 2.8 \), and \( \mu = 68.6 \): \[ X = -0.25 \cdot 2.8 + 68.6 = -0.7 + 68.6 = 67.9 \text{ in} \]So, a doorway height of 67.9 in. would be required for 40% of men to fit without bending.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-score calculation
A Z-score helps us understand how far a particular value is from the mean of a dataset, measured in terms of standard deviations. The formula to calculate the Z-score is:
mean and standard deviation
The two key parameters for normal distribution analysis are mean (
probability distribution analysis
A probability distribution describes how the values of a random variable are distributed. In the context of normal distribution, this means understanding how the data is spread around the mean.

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