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.) $$\begin{array}{|l|l|l|l|} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \mathrm{in} . & 1.1 \mathrm{in} . & \text { Normal } \\ \hline \text { Females } & 22.7 \mathrm{in} . & 1.0 \mathrm{in} . & \text { Normal } \\ \hline \end{array}$$ Find the probability that a male has a back-to-knee length less than 21 in.

Step by step solution

01

Define the Problem

Given the mean \((\mu)\) and standard deviation \((\sigma)\) for back-to-knee length in males, and knowing the data follows a normal distribution, determine the probability that a male's back-to-knee length is less than 21 inches.

02

Identify the Given Values

Extract the necessary values from the table.$$\mu_{males} = 23.5 \mathrm{in}$$ and$$\sigma_{males}=1.1 \mathrm{in}.$$

03

Standardize the Value

Calculate the Z-score for 21 inches using the formula \(Z = \frac{X-\mu}{\sigma}\). Substituting the values, we get$$Z = \frac{21-23.5}{1.1}.$$

04

Simplify the Z-Score

Perform the arithmetic to find the Z-score.$$Z = \frac{21-23.5}{1.1} = \frac{-2.5}{1.1} \approx -2.27.$$

05

Find the Cumulative Probability

Use the Z-score table to find the cumulative probability for Z = -2.27. The corresponding probability is approximately 0.0116.

06

Interpret the Result

The probability that a male has a back-to-knee length of less than 21 inches is about 0.0116 or 1.16%.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

probability

Probability is a measure of how likely an event is to occur. It ranges from 0 to 1, with 0 meaning the event will not happen and 1 meaning it will certainly happen.
In the context of our exercise, we want to understand how likely it is that a male’s back-to-knee length will be less than 21 inches.
This involves calculating the area under the curve of the normal distribution for the given data.
By converting measurements to Z-scores, we can use standard statistical tables or tools to find this probability.

normal distribution

A normal distribution, also known as a Gaussian distribution, is a bell-shaped curve that is symmetrical around the mean.
Most values will cluster around the mean, with fewer as they move away.
Important characteristics of a normal distribution include the mean (central peak) and standard deviation (spread).
In our exercise, the back-to-knee lengths of males and females are normally distributed.
This allows us to use the properties of normal distribution to calculate probabilities.

Z-score

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values.
It indicates how many standard deviations an element is from the mean.

• Z-score = (X - μ) / σ

In this formula, X is the value we are focusing on, is the mean, and σ is the standard deviation.
For our example, we calculate the Z-score to know how many standard deviations the back-to-knee length of 21 inches is from the mean of 23.5 inches for males.
This standardization allows us to use statistical tables to find probabilities.

cumulative probability

Cumulative probability refers to the probability that a random variable is less than or equal to a particular value.
When we calculate the Z-score and then refer to a standard normal distribution table, we find the cumulative probability.
This tells us the likelihood that a randomly selected male will have a back-to-knee length less than 21 inches.
In our step-by-step solution, we found the cumulative probability to be approximately 0.0116, meaning there is a 1.16% chance of this occurring.

mean and standard deviation

The mean (μ) and standard deviation (σ) are crucial statistics for understanding normal distributions.

• The mean is the average value, representing the center of the data.
• The standard deviation measures how spread out the values are from the mean.

In our exercise, the mean back-to-knee length for males is 23.5 inches and the standard deviation is 1.1 inches.
This information helps us understand the distribution of measurements and calculate probabilities.
When data follows a normal distribution, knowing these two values allows us to use Z-scores to find probabilities for any given value.