Question

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 21.6 in. and a standard deviation of 1.1 in. females sitting knee heights that are normally distributed with a mean of 19.1 in. and a standard deviation of 1.0 in. use this information to answer the following questions.
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 true 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 the 95th percentile for men is greater than the 5th percentile for women.
d. the statement is false because some women will have sitting knee heights that are outliers.
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?
95.82% (round to two decimal places as needed.)
what percentage of women fit this table?
100.00% (round to two decimal places as needed.)
does the table appear to be made to fit almost everyone? choose the correct answer below.
a. the table will fit only 4% of men.
b. the table will fit only 1% of women.
c. the table will fit almost everyone except about 4% of men with the largest sitting knee heights.
d. not enough information to determine if the table appears to be made to fit almost everyone.

Explanation:

Step1: Analyze the first - statement

The 95th percentile for men is greater than the 5th percentile for women. If there is clearance for 95% of males (which means the height is at or above the 95th - percentile male sitting knee height), it will be above the 5th - percentile female sitting knee height. So, the statement "If there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%" is true.

Step2: Calculate the percentage of men that fit the table

We use the z - score formula $z=\frac{x - \mu}{\sigma}$, where $x = 23.5$, $\mu=21.6$, and $\sigma = 1.1$. So, $z=\frac{23.5 - 21.6}{1.1}=\frac{1.9}{1.1}\approx1.73$. Looking up the z - value in the standard normal distribution table, the area to the left of $z = 1.73$ is approximately $0.9582$ or $95.82\%$.

Step3: Calculate the percentage of women that fit the table

For women, $\mu = 19.1$ and $\sigma = 1.0$. The z - score for $x = 23.5$ is $z=\frac{23.5 - 19.1}{1.0}=4.4$. The area to the left of $z = 4.4$ in the standard - normal distribution is very close to 1 or $100.00\%$.

Step4: Determine if the table fits almost everyone

Since about $95.82\%$ of men and $100\%$ of women fit the table, the table will fit almost everyone except about $(1 - 0.9582)\times100 = 4.18\%\approx4\%$ of men with the largest sitting knee heights.

Answer:

1. C. The statement is true because the 95th percentile for men is greater than the 5th percentile for women
2. $95.82\%$
3. $100.00\%$
4. C. The table will fit almost everyone except about 4% of men with the largest sitting knee heights