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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 the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Find the probability that a female has a back-to-knee length greater than \(24.0 \mathrm{in}\).

After 1964 , quarters were manufactured so that their weights have a mean of \(5.67 \mathrm{~g}\) and a standard deviation of \(0.06 \mathrm{~g}\). Some vending machines are designed so that you can adjust the weights of quarters that are accepted. If many counterfeit coins are found, you can narrow the range of acceptable weights with the effect that most counterfeit coins are rejected along with some legitimate quarters. a. If you adjust your vending machines to accept weights between \(5.60 \mathrm{~g}\) and \(5.74 \mathrm{~g}\), what percentage of legal quarters are rejected? Is that percentage too high? b. If you adjust vending machines to accept all legal quarters except those with weights in the top \(2.5 \%\) and the bottom \(2.5 \%\), what are the limits of the weights that are accepted?

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?

Use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution. A sample of depths (km) of earthquakes is obtained from Data Set 21 "Earthquakes" in Appendix B: 17.3, 7.0,7.0,7.0,8.1,6.8.

A normal distribution is informally described as a probability distribution that is bell-shaped when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.
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