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A common design requirement is that an environment must fit the range of people who fall between the Sth percentile for women and the 95 th percentile for men. In designing an assembly work table, we must consider sitting knee height, which is the distance from the bottom of the feet to the top of the knee. Males have sitting knee heights that are normally distributed with a mean of 21.4 in. and a standard deviation of 1.2 in.; females have sitting knee heights that are normally distributed with a mean of 19.6 in. and a standard deviation of 1.1 in. (based on data from the Department of Transportation). a. What is the minimum table clearance required to satisfy the requirement of fitting \(95 \%\) of men? Why is the 95 th percentile for women ignored in this case? b. The author is writing this exercise at a table with a clearance of 23.5 in. above the floor. What percentage of men fit this table, and what percentage of women fit this table? Does the table appear to be made to fit almost everyone?

Step by step solution

01

- Understand the problem

We need to determine the minimum table clearance required to fit 95% of men based on their sitting knee height and also determine what percentage of men and women can fit under a table with a clearance of 23.5 inches.

02

- Calculate the 95th percentile for men

The sitting knee height for men is normally distributed with a mean (\u03bc) of 21.4 inches and a standard deviation (\u03c3) of 1.2 inches. To find the 95th percentile, we use the z-score that corresponds to 0.95 in the standard normal distribution table, which is approximately 1.645. The 95th percentile for men is given by \[ \text{95th percentile} = \u03bc + (z \times \u03c3) \]Plugging in the values, we get \[ 21.4 + (1.645 \times 1.2) = 21.4 + 1.974 = 23.374 \text{ inches} \]

03

- Explain why the 95th percentile for women is ignored

The 95th percentile for women is ignored because it is unnecessary for the table design. The table is designed to accommodate the largest possible sitting knee height, which corresponds to the 95th percentile of men. Women's sitting knee heights are generally lower, so ensuring the table fits 95% of men will automatically accommodate virtually all women.

04

- Calculate percentage of men who can fit a 23.5-inch table

We need to find what percentage of men have a sitting knee height less than 23.5 inches. Using the z-score formula:\[ z = \frac{X - \mu}{\sigma} \]Where X is 23.5, \mu is 21.4, and \sigma is 1.2, we get\[ z = \frac{23.5 - 21.4}{1.2} = \frac{2.1}{1.2} = 1.75 \]<|vq_3763|>

05

- Use the z-score table for percentage of men

Refer to the z-score table to determine the percentage corresponding to a z-score of 1.75, which is approximately 0.9599, or 95.99%. So, 95.99% of men can fit under a table with a 23.5-inch clearance.

06

- Calculate percentage of women who can fit a 23.5-inch table

Using the same z-score formula for women, where \mu is 19.6 and \sigma is 1.1, we get\[ z = \frac{23.5 - 19.6}{1.1} = \frac{3.9}{1.1} = 3.545 \]The z-score of 3.545 corresponds to a percentage close to 100%. Thus, virtually all women can fit under the table.

07

- Conclusion on table design

Given that 95.99% of men and virtually all women can fit under the table with a clearance of 23.5 inches, the table appears to be designed to fit almost everyone.

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Key Concepts

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

normal distribution

In statistics, a normal distribution is a common continuous probability distribution. The curve of a normal distribution is bell-shaped and symmetric about the mean. It is often used in statistical analysis because many human characteristics, such as height or sitting knee height, naturally follow this pattern. When data is normally distributed, most people will have values around the average (mean) value, with fewer people having values far from the mean. For instance, in the problem, both male and female sitting knee heights follow a normal distribution. This means most men will have knee heights around 21.4 inches and most women around 19.6 inches.

z-score

A z-score, or standard score, indicates how many standard deviations an element is from the mean. The formula to calculate the z-score is: \[z = \frac{X - \mu}{\sigma}\] Where:

• X is the value you are examining,
• \( \mu \) is the mean, and,
• \(\sigma\) is the standard deviation.

For example, to find how far a sitting knee height of 23.5 inches is from the mean for men, we calculate the z-score. If the mean knee height is 21.4 inches with a standard deviation of 1.2 inches, the z-score is: \[z = \frac{23.5 - 21.4}{1.2} \approx 1.75\] This means that 23.5 inches is 1.75 standard deviations above the mean.

statistical analysis

Statistical analysis involves collecting, analyzing, interpreting, presenting, and organizing data. It helps us understand patterns and trends in data. In the given problem, statistical analysis is used to understand the range of sitting knee heights for men and women. We used the mean and standard deviation values to perform calculations and determine the necessary table clearance. We also used the normal distribution and z-scores to find the percentiles and understand what percentage of people can fit under a table of specific height.

percentile rank

The percentile rank indicates the relative standing of a value in a data set. For example, the 95th percentile means that 95% of the values are below that point and 5% are above it. To calculate the 95th percentile for men's sitting knee heights, we use the mean and standard deviation: \[\text{95th percentile} = \mu + (z \times \sigma)\] For men, with a mean of 21.4 inches and standard deviation of 1.2 inches:\[21.4 + (1.645 \times 1.2) \approx 23.374\] Thus, at least 95% of men will have a sitting knee height below 23.374 inches. Percentile ranks help in making design decisions, like determining the minimum table clearance required to fit a specific range of people.