91Ó°ÊÓ

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.) $$\begin{array}{|l|l|l|l|} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \mathrm{in} . & 1.1 \mathrm{in} . & \text { Normal } \\ \hline \text { Females } & 22.7 \mathrm{in} . & 1.0 \mathrm{in} . & \text { Normal } \\ \hline \end{array}$$ For males, find \(P_{90}\) which is the length separating the bottom \(90 \%\) from the top \(10 \%\)

Step by step solution

01

Understand what is given

Identify the given values and parameters from the problem. For males, the mean length is 23.5 inches and the standard deviation is 1.1 inches.

02

Understand the meaning of P_90

P_90 represents the 90th percentile. This means that 90% of the values lie below this length, and 10% lie above it.

03

Use Standard Normal Distribution

Transform the given parameters into the standard normal distribution (mean = 0, standard deviation = 1) by using the z-score formula.

04

Set up the z-score formula

The z-score formula is: \[ z = \frac{X - \text{mean}}{\text{standard deviation}} \] Where X is the value corresponding to P_90.

05

Find the z-score for the 90th percentile

Using standard normal distribution tables or a z-score calculator, find the z-score that corresponds to the 90th percentile. The z-score for P_90 is approximately 1.28.

06

Solve for X using the z-score formula

Plug the z-score and the given parameters into the z-score formula to solve for X: \[ X = \text{mean} + (z \times \text{standard deviation}) \] \[ X = 23.5 + (1.28 \times 1.1) \] \[ X = 23.5 + 1.408 \] \[ X = 24.908 \]

07

Interpret the result

The value of P_90 represents the length below which 90% of the male population sits. For males, P_90 is approximately 24.91 inches.

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Key Concepts

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

normal distribution

The normal distribution is a crucial concept in statistics and probability. It is also known as the Gaussian distribution. This type of distribution is symmetrical, with most data points clustering around the mean. As you move further away from the mean, the frequency of data points decreases.

Features of normal distribution include:

• Symmetrical shape
• Bell curve
• Mean, median, and mode are all equal

In our exercise, the data for both males and females follow a normal distribution, allowing us to use specific statistical methods like z-scores to find percentiles. Understanding normal distribution is vital for interpreting various types of data correctly.

z-score

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean. For example, a z-score of 1.28 means the value is 1.28 standard deviations above the mean.

Steps to calculate a z-score:

• Identify the mean and standard deviation of the dataset.
• Use the formula: \[ z = \frac{X - \text{mean}}{\text{standard deviation}} \]

In our example, we used the z-score formula to convert the given mean and standard deviation of male lengths into a standard normal distribution. This enabled us to find the value at the 90th percentile, by knowing the z-score corresponds to that percentile.

standard deviation

Standard deviation measures the dispersion of a dataset relative to its mean. A high standard deviation indicates that the data points are spread out over a large range of values, while a low standard deviation indicates that they are close to the mean.

To calculate standard deviation:

• Find the mean of the dataset.
• Subtract the mean from each data point and square the result.
• Find the average of these squared differences.
• Take the square root of this average.

In the exercise, the standard deviation for sitting lengths was provided, which allowed us to use it with the mean and z-score to solve for the 90th percentile.

mean

The mean, often referred to as the average, is a central value of a finite set of numbers. It is found by dividing the sum of all data points by the number of points. The mean represents a typical value in the set.

Mean calculation steps:

• Add together all the values in the dataset.
• Divide the sum by the number of values.

In our example, the mean sitting lengths for males and females were given directly. This mean value was utilized in calculating the z-score and eventually finding the 90th percentile, which helps in understanding how the values are distributed around this central value.