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 21.4 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. the author is writing this exercise at a table with a clearance of 23.3 in. above the floor. what percentage of men fit this table? % (round to two decimal places as needed.)

Explanation:

Step1: Recall the z - score for 95th percentile

For a normal distribution, the z - score corresponding to the 95th percentile is approximately $z = 1.645$.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean and $\sigma$ is the standard deviation. We know that for men, $\mu = 21.4$ in, $\sigma=1.2$ in and $z = 1.645$. We want to find $x$. Rearranging the formula gives $x=\mu + z\sigma$.

Step3: Calculate the 95th percentile for men

Substitute the values into the formula: $x=21.4+1.645\times1.2=21.4 + 1.974=23.4$ in.

For the second part:
The 95th percentile for men is the upper - bound for 95% of men's sitting knee heights. The 5th percentile for women is the lower - bound for 95% of women's sitting knee heights. Since the 95th percentile for men is greater than the 5th percentile for women, if there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%.

For the third part:

Step1: Use the z - score formula again

We have $\mu = 21.4$ in, $\sigma = 1.2$ in and $x = 23.3$ in. Calculate the z - score $z=\frac{x-\mu}{\sigma}=\frac{23.3 - 21.4}{1.2}=\frac{1.9}{1.2}\approx1.583$.

Step2: Find the percentile using the z - score

We look up the z - score of 1.583 in the standard normal distribution table. The area to the left of $z = 1.583$ is approximately 0.943. So the percentage of men that fit the table is 94.30%.

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
3. 94.30%