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 heights have to be considered, which is the distance from the bottom of the foot 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 19.1 in and a standard deviation of 1.1 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?
23.4 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 the 95th percentile for men is greater than the 5th percentile for women
b. the statement is false because some women will have sitting knee heights that are outliers.
c. the statement is true because the 95th percentile for men is greater than the 5th percentile for women.
d. the statement is true because some women will have sitting knee heights that are outliers.

Explanation:

Step1: Recall normal - distribution property

We know that if we want to find the value corresponding to a certain percentile in a normal distribution, we use the z - score formula $z=\frac{x-\mu}{\sigma}$, where $x$ is the value we want to find, $\mu$ is the mean, $\sigma$ is the standard deviation, and $z$ is the z - score from the standard normal distribution table.
For men, $\mu = 21.4$ in and $\sigma=1.2$ in. We want to find the 95th percentile of men's sitting knee heights. Looking up the z - score in the standard normal distribution table for the 95th percentile, $z = 1.645$.

Step2: Solve for $x$ (sitting knee height)

Using the z - score formula $z=\frac{x-\mu}{\sigma}$, we can re - arrange it to $x=\mu + z\sigma$. Substituting the values $\mu = 21.4$, $z = 1.645$, and $\sigma = 1.2$ into the formula:
$x=21.4+1.645\times1.2=21.4 + 1.974=23.374\approx23.4$ in.

For the second part:
The 95th percentile of men's sitting knee heights is greater than the 5th percentile of women's sitting knee heights. If the table clearance is enough for the 95th percentile of men, it will be enough for the 5th percentile of women.
The mean of women's sitting knee heights is 19.1 in with a standard deviation of 1.1 in. The 5th percentile of women's sitting knee heights has a z - score of approximately $z=- 1.645$. Using $x=\mu+z\sigma$, we get $x = 19.1+( - 1.645)\times1.1=19.1-1.8095 = 17.2905$ in. Since the 95th percentile of men ($\approx23.4$ in) is greater than the 5th percentile of women, if there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%.

Answer:

1. 23.4 in
2. C. The statement is true because the 95th percentile for men is greater than the 5th percentile for women.