help! statistics (normal distribution)

If a car seat is too low to too high, it will be uncomfortable and possibly dangerous. Most seats are made to be adjustable, so that men and women with different sitting heights can select a comfortable setting. When designing car seats, the sitting knee heights of men and women become very relevant. Men have sitting knee heights that are normally distributed with a mean of 22.0 in. and a standard deviation of 1.1 in.., and women have sitting knee heights that are normally distributed with a mean of 20.3 in. and a standard deviation of 1.0 in. (based on anthropometric survey data from Gordon, Churchill etc. al)


A) Find minimum and maximum sitting knee heights that include at least 95% of all men and at least 95% of all women.. Remember, at least 95% or more.

B) Using your answer from part (a), try to find cost-effective limits that are close together as possible. Remember, there are no separate seats designed for men or women You have to decide what minimum and maximum sitting knee heights you will be using, i.e. choose one minimum and one maximum sitting knee height. Clearly explain your reasoning for choosing your minimum and maximum sitting knee heights.

C) Using the maximum and minimum sitting knee heights from part (b), find the percentage of men with sitting knee heights between the limits you have determined, and then find the percentage of women with sitting knee heights between those limits. Do your limits favor one gender at the expense of the other? Why or Why not? Explain briefly.

D) Why is it not practical to simply design car seats so that they accommodate everyone?

E) If you were a design engineer for General Motors, what percentage would you be willing to exclude in your design of car seats and why? Explain

F) Apart from sitting knee height, what is another important design component that should be considered when determining the adjustment range or car seats?

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First I must apologize for not knowing LaTeX.

A)

For men and women separately we have:

Women:

Let X be the knee height of women.
Then X ~ Normal(20.3, (1.0)^2)
Then (X-20.3)/(1.0) ~ Normal(0,1)
Now we find A such that P(X>A) = .95

-> P(Normal(0,1) > (A-20.3)/1.0) = .95

Take the inverse normal with mean 0 and st. dev 1 and you find:

1.64485 > (A-20.3)/1.0)
-> A < 20.3 + 1.64485 = 21.94485

This is the value such that 95% of women have sitting knee heights below this value.

A similar line of reasoning gives:

B > 20.3 - 1.64485 = 18.65515

This is the value such that 95% of women have a sitting knee height above this value.

You can also find the value C such that 95% of women have a knee height in the range 20.3 +- C.

In this case you would take the inverse normal with area .975, mean 0, and st. dev 1, which gives:

C = 1.95996

-> 1.95996 > (A - 20.3)/1.0 > -1.95996

-> 22.25996 > A > 18.34004

This implies that 95% of women have a knee height between these two values.

Men:

Follow the same steps as with women:

X ~ Normal(22.0, (1.1)^2)

You'll find 95% of men have a knee height below 23.809 in., 95% have a knee height above 20.190 in., and 95% have a knee height between 19.844 and 24.15596 in.

B)

Now, if you want to include all men and women, you might take the minimum height of 18.65 in. (min for women) and a max height of 23.81 in. (max for men). This way you can accommodate 95% of women and more than 95% of men on the lower end, and 95% of men and more than 95% of women on the upper end. Notice that this does not necessarily ensure 95% of women and 95% of men are accommodated since we assumed upper and lower limits of infinity when finding the minimum and maximum respectively. We will check this out in part C.


C) Now we use the normal CDF for women and men across this range.

For women we take the CDF value at 23.81 minus the value at 18.65 using a mean of 20.3 and st. dev of 1.0 giving:

.9503

Meaning 95.03% of women have knee heights between 18.65 and 23.81 inches, looks good.

For men we do the same except mean is 22.0 and st. dev is 1.1:

.9489

Meaning 94.89% of men have knee heights between 18.65 and 23.81 inches. Notice this is almost but not quite 95%.

This seems to indicate that in our range women are slightly favored. One could change up the range a bit if they wish, but this looks good to me personally.

D) Obviously you cannot accommodate everyone. Someone with enormously long shins will have a problem. When it comes to an exact normal distribution, you'd have to have your range go from 0 to infinity to accommodate "100%," which is obviously impossible.



I hope this helped a little? Maybe not...

Quote:

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Originally Posted by hjwach

If a car seat is too low to too high, it will be uncomfortable and possibly dangerous. Most seats are made to be adjustable, so that men and women with different sitting heights can select a comfortable setting. When designing car seats, the sitting knee heights of men and women become very relevant. Men have sitting knee heights that are normally distributed with a mean of 22.0 in. and a standard deviation of 1.1 in.., and women have sitting knee heights that are normally distributed with a mean of 20.3 in. and a standard deviation of 1.0 in. (based on anthropometric survey data from Gordon, Churchill etc. al)


A) Find minimum and maximum sitting knee heights that include at least 95% of all men and at least 95% of all women.. Remember, at least 95% or more.

B) Using your answer from part (a), try to find cost-effective limits that are close together as possible. Remember, there are no separate seats designed for men or women You have to decide what minimum and maximum sitting knee heights you will be using, i.e. choose one minimum and one maximum sitting knee height. Clearly explain your reasoning for choosing your minimum and maximum sitting knee heights.

C) Using the maximum and minimum sitting knee heights from part (b), find the percentage of men with sitting knee heights between the limits you have determined, and then find the percentage of women with sitting knee heights between those limits. Do your limits favor one gender at the expense of the other? Why or Why not? Explain briefly.

D) Why is it not practical to simply design car seats so that they accommodate everyone?

E) If you were a design engineer for General Motors, what percentage would you be willing to exclude in your design of car seats and why? Explain

F) Apart from sitting knee height, what is another important design component that should be considered when determining the adjustment range or car seats?

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A thread of related interest: http://www.mathhelpforum.com/math-he...e-heights.html