Questions: A common design requirement is that an environment must fit the range of people who fall between the 5th percentile for women and the 95th percentile for men. In designing an assembly work table, the sitting knee height must be considered, 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 21.3 in. and a standard deviation of 1.3 in. Females have sitting knee heights that are normally distributed with a mean of 19.4 in. and a standard deviation of 1.2 in. Use this information to answer the following questions. What is the minimum table clearance required to satisfy the requirement of fitting 95% of men? in. (Round to one decimal place as needed.) Determine if the following statement is true or false. If there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%. A. The statement is false because some women will have sitting knee heights that are outliers. B. The statement is false because the 95th percentile for men is greater than the 5th percentile for women. C. The statement is true because some women will have sitting knee heights that are outliers. D. The statement is true because the 95th percentile for men is greater than the 5th percentile for women. The author is writing this exercise at a table with a clearance of 23.7 in. above the floor. What percentage of men fit this table? % (Round to two decimal places as needed.) What percentage of women fit this table?

A common design requirement is that an environment must fit the range of people who fall between the 5th percentile for women and the 95th percentile for men. In designing an assembly work table, the sitting knee height must be considered, 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 21.3 in. and a standard deviation of 1.3 in. Females have sitting knee heights that are normally distributed with a mean of 19.4 in. and a standard deviation of 1.2 in. Use this information to answer the following questions. What is the minimum table clearance required to satisfy the requirement of fitting 95% of men? in. (Round to one decimal place as needed.) Determine if the following statement is true or false. If there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%. A. The statement is false because some women will have sitting knee heights that are outliers. B. The statement is false because the 95th percentile for men is greater than the 5th percentile for women. C. The statement is true because some women will have sitting knee heights that are outliers. D. The statement is true because the 95th percentile for men is greater than the 5th percentile for women. The author is writing this exercise at a table with a clearance of 23.7 in. above the floor. What percentage of men fit this table? % (Round to two decimal places as needed.) What percentage of women fit this table?
Transcript text: A common design requirement is that an environment must fit the range of people who fall between the $5^{\text {th }}$ percentile for women and the $95^{\text {th }}$ percentile for men. In designing an assembly work table, the sitting knee height must. be considered, which is the distance from the bottom of the feet to the top of the knee. Males have sitting knee height: that are normally distributed with a mean of 21.3 in. and a standard deviation of 1.3 in . Females have sitting knee heights that are normally distributed with a mean of 19.4 in. and a standard deviation of 1.2 in. Use this information to answer the following questions. What is the minimum table clearance required to satisfy the requirement of fitting $95 \%$ of men? $\square$ in. (Round to one decimal place as needed.) Determine if the following statement is true or false. If there is clearance for $95 \%$ of males, there will certainly be clearance for all women in the bottom $5 \%$. A. The statement is false because some women will have sitting knee heights that are outliers. B. The statement is false because the 95 th percentile for men is greater than the 5 th percentile for women. C. The statement is true because some women will have sitting knee heights that are outliers. D. The statement is true because the 95 th percentile for men is greater than the 5 th percentile for women. The author is writing this exercise at a table with a clearance of 23.7 in . above the floor. What percentage of men fit this table? $\square$ \% (Round to two decimal places as needed.) What percentage of women fit this table?
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Solution

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Solution Steps

To solve the given questions, we need to use properties of the normal distribution. Specifically, we will use the mean and standard deviation to find the required percentiles and probabilities.

  1. Minimum table clearance for 95% of men:

    • Use the mean and standard deviation of men's sitting knee height to find the 95th percentile.

  2. Determine the truth of the statement:

    • Compare the 95th percentile of men's sitting knee height with the 5th percentile of women's sitting knee height.

  3. Percentage of men fitting a table with 23.7 in clearance:

    • Use the cumulative distribution function (CDF) to find the percentage of men with sitting knee height less than or equal to 23.7 in.

  4. Percentage of women fitting a table with 23.7 in clearance:

    • Use the CDF to find the percentage of women with sitting knee height less than or equal to 23.7 in.
Step 1: Minimum Table Clearance for 95% of Men

To find the minimum table clearance required to accommodate 95% of men, we calculate the 95th percentile of the normal distribution for men's sitting knee height, which is given by:

\[ X_{95} = \mu + z \cdot \sigma \]

where \( \mu = 21.3 \) in, \( \sigma = 1.3 \) in, and \( z \) for the 95th percentile is approximately \( 1.645 \). The calculated value is:

\[ X_{95} \approx 23.4383 \text{ in} \]

Rounding to one decimal place, the minimum table clearance is:

\[ \boxed{23.4 \text{ in}} \]

Step 2: Truth of the Statement

We need to evaluate the statement: "If there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%."

We compare the 95th percentile for men with the 5th percentile for women:

  • 95th percentile for men: \( X_{95} \approx 23.4383 \) in
  • 5th percentile for women: \( X_{5} \approx 17.4262 \) in

Since \( 23.4383 > 17.4262 \), the statement is true because the 95th percentile for men is greater than the 5th percentile for women.

Thus, the answer is:

\[ \text{The statement is true because the 95th percentile for men is greater than the 5th percentile for women.} \]

Step 3: Percentage of Men Fitting the Table

To find the percentage of men that fit a table with a clearance of 23.7 in, we calculate the cumulative distribution function (CDF):

\[ P(X \leq 23.7) = \Phi\left(\frac{23.7 - \mu}{\sigma}\right) \]

Substituting the values:

\[ P(X \leq 23.7) \approx 96.7565\% \]

Rounding to two decimal places, the percentage of men fitting the table is:

\[ \boxed{96.76\%} \]

Step 4: Percentage of Women Fitting the Table

Similarly, for women, we calculate the percentage fitting the same table clearance:

\[ P(X \leq 23.7) = \Phi\left(\frac{23.7 - 19.4}{1.2}\right) \]

This results in:

\[ P(X \leq 23.7) \approx 99.9830\% \]

Rounding to two decimal places, the percentage of women fitting the table is:

\[ \boxed{99.98\%} \]

Final Answer

  • Minimum table clearance for 95% of men: \(\boxed{23.4 \text{ in}}\)
  • Truth of the statement: The statement is true.
  • Percentage of men fitting the table: \(\boxed{96.76\%}\)
  • Percentage of women fitting the table: \(\boxed{99.98\%}\)
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