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

21.9

in. and a standard deviation of

1.2

in. Females have sitting knee heights that are normally distributed with a mean of

19.2

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? (round to one decimal 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 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 false 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

23.5

in. above the floor. What percentage of men fit this​ table?

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? (round to two decimals)

What percentage of women fit this​ table? (round to two decimals)

Does the table appear to be made to fit almost​ everyone? Choose the correct answer below.

A.

The table will only fit​ 1% of women.

B.The table will fit only

99​%

of men.

C.The table will fit almost everyone except about

99​%

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.

Answer 1

Let x be the sitting knee heights of the women and y be the sitting knee heights of the men

x follows normal distribution with a mean µ = 19.2 in. and a standard deviation σ = 1.1

y follows normal distribution with a mean µ = 21.9 in. and a standard deviation σ = 1.2

#1) What is the minimum table clearance required to satisfy the requirement of fitting​ 95% of​ men?

We have to find y such that area to the left is 0.95

So first we need to find z score corresponding to area 0.95 , such z score is 1.645 from the z score table.

Therefore y = z*σ + µ  

= (1.645*1.2) + 21.9

= 23.9

#2)

If there is clearance for​ 95% of​ males, there will certainly be clearance for all women in the bottom​ 5%.

D. The statement is true because the 95^th percentile for men is greater than the 5th percentile for women.

#3)

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?

P( y < 23.5)

=

= P( z < 1.33)

= 0.9082 ----( from z score table , table value corresponding to z = 1.33 )

So 90.82% of men fit this​ table.

#4) What percentage of women fit this​ table?

P( x < 23.5)

=

= P( z < 3.91)

= 0.9999 ------ ( from z score table , table value corresponding to z = 3.91 )

So 99.99% of women fit this​ table.

#5) Does the table appear to be made to fit almost​ everyone?

C.The table will fit almost everyone except about 99​% of men with the largest sitting knee heights.