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Chapter 6: Problem 21

Use these parameters (based on Data Set 1 "Body Data" in Appendix B): \- Men's heights are normally distributed with mean \(68.6 \mathrm{in.}\) and standard deviation \(2.8 \mathrm{in} .\) \- Women's heights are normally distributed with mean 63.7 in. and standard deviation \(2.9 \mathrm{in}\). The U.S. Navy requires that fighter pilots have heights between 62 in. and 78 in. a. Find the percentage of women meeting the height requirement. Are many women not qualified because they are too short or too tall? b. If the Navy changes the height requirements so that all women are eligible except the shortest \(3 \%\) and the tallest \(3 \%\), what are the new height requirements for women?

Short Answer

Expert verified
a. 72.2% of women meet the requirement. b. New height range: 58.174 in. to 69.226 in.

Step by step solution

01

- Understand the Problem

Identify the parameters provided: Men's mean height 68.6 in. (SD 2.8 in.), Women's mean height 63.7 in. (SD 2.9 in.). The height requirement for Navy fighter pilots is between 62 in. and 78 in.
02

- Standardize Women's Heights with Z-Scores

Calculate the Z-scores for the heights 62 in. and 78 in. using the formula: \( Z = \frac{X - \mu}{\sigma} \) \(Z_{62} = \frac{62 - 63.7}{2.9} = -0.586\) \(Z_{78} = \frac{78 - 63.7}{2.9} = 4.931\)
03

- Find the Percentage of Women Meeting Height Requirements

Using the Z-table, find the proportions corresponding to the Z-scores: \(Z_{62} = -0.586 \implies 0.278\), \(Z_{78} = 4.931 \implies 1\). The percentage of women meeting the height requirement is: \((1 - 0.278) * 100\% = 72.2\%\)
04

- Interpret the Women's Height Data

72.2% of women meet the height requirement, meaning 27.8% do not meet the requirement due to being either too short or too tall.
05

- Determine the New Height Requirements for Women

Find the Z-scores corresponding to the shortest and tallest 3%: for the lower 3%, \(Z_{0.03} = -1.88\), and for the upper 3%, \(Z_{0.97} = 1.88\).
06

- Convert Z-Scores Back to Heights

Convert the Z-scores back to heights using the formula: \(X = \mu + Z \cdot \sigma\): \(X_{min} = 63.7 + (-1.88) \cdot 2.9 = 58.174\) \(X_{max} = 63.7 + 1.88 \cdot 2.9 = 69.226\)

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Key Concepts

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

Normal Distribution
In statistics, many types of data can be represented using the normal distribution, also known as the bell curve because of its shape. This curve is symmetrical, with most of the data clustering around the mean (average) value. The curve tails off equally on both sides, representing fewer occurrences of values farther from the mean. Knowing that a dataset is normally distributed helps us make future predictions and calculate probabilities.
Z-Scores
A Z-score helps us understand how far away a particular value is from the mean, measured in terms of standard deviations. The formula for calculating a Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
  • X is the value being evaluated
  • \( \mu \) (mu) is the mean of the data
  • \( \sigma \) (sigma) is the standard deviation

By standardizing scores, we can compare different datasets directly and use Z-tables to find probabilities associated with any given value.
Percentage Calculation
Percentage calculation in this context helps us determine the proportion of individuals meeting certain criteria within a normal distribution. Once we have our Z-scores, we can refer to a Z-table to find corresponding proportions. For example, if the Z-score for a height of 62 inches for women is -0.586, the Z-table tells us that approximately 27.8% of women have a height below this value. Therefore, the percentage of women within the specified height range is calculated by:

\[ \text{Percentage} = (1 - 0.278) \times 100% = 72.2% \]
This means 72.2% of women fulfill the height requirement.
Height Requirements
Height requirements are often used in various settings such as the military, sports, and general health guidelines. When determining if someone meets specific height criteria, we consider the distribution of heights within a population. In the Navy's case, they set a range for acceptable pilot heights, initially between 62 inches and 78 inches. Further refining these requirements to exclude the shortest and tallest 3% means finding new Z-scores, such as
\(Z_{0.03} = -1.88 \) and
\( Z_{0.97} = 1.88 \).

Converting these back to real height values, we find:
\[X_{min} = 63.7 + (-1.88) \times 2.9 \approx 58.2 \text{ in}\]
\[X_{max} = 63.7 + 1.88 \times 2.9 \approx 69.2 \text{ in}\]
This ensures all eligible women fall between these new heights.

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Most popular questions from this chapter

Use these parameters (based on Data Set 1 "Body Data" in Appendix B): \- Men's heights are normally distributed with mean \(68.6 \mathrm{in.}\) and standard deviation \(2.8 \mathrm{in} .\) \- Women's heights are normally distributed with mean 63.7 in. and standard deviation \(2.9 \mathrm{in}\). Air Force Pilots The U.S. Air Force requires that pilots have heights between 64 in. and 77 in. a. Find the percentage of men meeting the height requirement. b. If the Air Force height requirements are changed to exclude only the tallest \(2.5 \%\) of men and the shortest \(2.5 \%\) of men, what are the new height requirements?

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About \(\quad \%\) of the area is between \(z=-3.5\) and \(z=3.5\) (or within \(3.5\) standard deviations of the mean).

The heights (in inches) of men listed in Data Set 1 "Body Data" in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population. a. If 2 inches is added to each height, are the new heights also normally distributed? b. If each height is converted from inches to centimeters, are the heights in centimeters also. normally distributed? c. Are the logarithms of normally distributed heights also normally distributed?

Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of \(12.00 \mathrm{oz}\) and a standard deviation of \(0.11\) oz. a. Find the probability that a single can of Coke has at least \(12.19 \mathrm{oz}\). b. The 36 cans of Coke in Data Set 26 "Cola Weights and Volumes" in Appendix \(\mathrm{B}\) have a mean of \(12.19\) oz. Find the probability that 36 random cans of Coke have a mean of at least \(12.19 \mathrm{oz}\) c. Given the result from part (b), is it reasonable to believe that the cans are actually filled with a mean equal to \(12.00\) oz? If the mean is not equal to \(12.00 \mathrm{oz}\), are consumers being cheated?

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About _______ \(\%\) of the area is between \(z=-3\) and \(z=3\) (or within 3 standard deviations of the mean).
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