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A common design requirement is that an environment must fit the range of people who fall between the 5 th 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

Calculate the 95th Percentile for Men's Sitting Knee Height

To find the 95th percentile for men's sitting knee height, use the formula for the percentile in a normal distribution: \(X = \text{mean} + Z \times \text{standard deviation}\) For the 95th percentile, Z is approximately 1.645. Therefore, \(X = 21.4 + 1.645 \times 1.2 = 23.374\) inches.

02

Explain the Reason for Ignoring Women's 95th Percentile

The 95th percentile for men is used instead of the 95th percentile for women because the maximum clearance required to fit 95% of the males will inherently cover all females as well, considering the data given that males tend to have larger sitting knee heights than females.

03

Calculate the Z-score for the Table Clearance for Men

To determine what percentage of men can fit under a table with clearance of 23.5 inches, calculate the Z-score using the formula: \(Z = \frac{X - \text{mean}}{\text{standard deviation}}\) For men: \(Z = \frac{23.5 - 21.4}{1.2} = 1.75\)

04

Find the Percentage of Men that Fit the Table

Using the Z-score table, a Z-score of 1.75 corresponds to approximately 95.97%. Therefore, approximately 95.97% of men can fit under a table with 23.5 inches clearance.

05

Calculate the Z-score for the Table Clearance for Women

To determine what percentage of women can fit under the same table, calculate the Z-score using the formula: \(Z = \frac{X - \text{mean}}{\text{standard deviation}}\) For women: \(Z = \frac{23.5 - 19.6}{1.1} = 3.545\)

06

Find the Percentage of Women that Fit the Table

Using the Z-score table, a Z-score of 3.545 corresponds to nearly 100%. Therefore, almost all women can fit under a table with 23.5 inches clearance.

07

Conclusion about the Table Design

Since 95.97% of men and nearly 100% of women can fit under the table with 23.5 inches clearance, the table is 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

When we talk about 'normal distribution', we are referring to a specific statistical concept where data points are symmetrically distributed around the mean. In simple terms, if you were to graph the distribution, it would form a bell-shaped curve. The highest point of the curve represents the mean, median, and mode of the data set.
Normal distribution is key in many fields, such as psychology, education, and engineering, because it allows for the standardization of values. This means that we can use the same approach to analyze different data sets that follow this pattern.
For example, in the problem, the men's sitting knee heights are normally distributed. This tells us that most men's sitting knee heights are close to the average (mean) of 21.4 inches, with fewer men having significantly longer or shorter sitting knee heights.

z-score calculation

Z-score calculation allows us to determine how far a specific data point is from the mean, measured in terms of standard deviations. The formula for calculating a Z-score is:
\[ Z = \frac{X - \text{mean}}{\text{standard deviation}} \]
Here, X is the data point, the mean is the average value, and the standard deviation is a measure of how spread out the values are around the mean.
In our exercise, to find what percentage of men can fit under a table 23.5 inches high, we calculated the Z-score for the table clearance:
\[ Z = \frac{23.5 - 21.4}{1.2} = 1.75 \]
This shows that 23.5 inches is 1.75 standard deviations above the mean. By consulting a Z-score table, we can find that a Z-score of 1.75 corresponds to about 95.97%, meaning that roughly 96% of men can fit under this table.

design requirements in statistics

Design requirements in statistics involve using statistical data to set criteria for products or environments to ensure they meet the needs of the intended users. This often involves considering percentiles to accommodate various populations.
For instance, in our assembly work table exercise, we consider the 5th percentile of women and the 95th percentile of men for knee height. By focusing on these percentiles, we can design tables that fit the vast majority of users. In this case, the 95th percentile for men is more critical as it includes even the larger heights, ensuring the table clearance is adequate.
This approach is data-driven and helps in creating inclusive designs that cater to nearly everyone in the target audience.

mean and standard deviation

Mean and standard deviation are fundamental concepts in statistics.
The 'mean' is the average value of a dataset. It's calculated by summing all the values and dividing by the number of values. For men鈥檚 sitting knee height, the mean is 21.4 inches.
The 'standard deviation' measures how spread out the numbers are in a dataset. A small standard deviation means most numbers are close to the mean, while a large standard deviation means the numbers are more dispersed. In our problem, the standard deviation for men is 1.2 inches, and for women, it is 1.1 inches.
Understanding these metrics is essential for interpreting data and making informed design decisions. In our exercise, knowing the mean and standard deviation allows us to calculate the required table clearance to accommodate a large percentage of both men and women.