Statistics Homework

Does anyone remember how to solve for x in this equation, I can't remember how to do it?

[10 (x)^9] - [9 (x)^10] = 0.75

The answer is 0.9036 but the prof does not say how to get it. Here is his only explanation:

The 75th percentile is the value of x for which
F(x) = 0.75
0.75 = 10(x)9 – 9(x)10
x » .9036

The equation does not seem invertible in x so I don't think you can solve it analytically.

What if you repeatedly take the derivative with respect to x until you have a quadratic? Will that work?

Originally posted by AThousandYoung
What if you repeatedly take the derivative with respect to x until you have a quadratic? Will that work?

I don't see why it should... but maybe I'm not following you. How would you go about it?

There is no analytical solution. There are two real roots and eight complex ones since this is a tenth-degree polynomial equation.

http://www.wolframalpha.com/input/?i=Solve%5B%5B10+%28x%29^9%5D+-+%5B9+%28x%29^10%5D+%3D+0.75%2Cx%5D

Originally posted by mlprior
Does anyone remember how to solve for x in this equation, I can't remember how to do it?

[10 (x)^9] - [9 (x)^10] = 0.75

The answer is 0.9036 but the prof does not say how to get it. Here is his only explanation:

The 75th percentile is the value of x for which
F(x) = 0.75
0.75 = 10(x)9 – 9(x)10
x » .9036

Im going with pylanka, just use some form of approximation method (I used Newtons method) to solve the polynomial. Since the powers are pretty large, to avoid a wild goose chase,I'd make sure you have a pretty accurate visual representation of the function in the range to get your first approximation.

P(x) = -9*x^10+10*x^9-.75 = 0

Recap of Newtons method:

x(n+1) = x(n) - f(x(n))/f'(x(n))

Originally posted by Palynka
I don't see why it should... but maybe I'm not following you. How would you go about it?

I wouldn't. I'm out of my depth here for now.

OK thanks.

I'm not going to worry about it then, it was just in one of the homework questions. I don't think the professor would actually put something like that on a test anyway.

Thanks everyone!

Originally posted by mlprior
OK thanks.

I'm not going to worry about it then, it was just in one of the homework questions. I don't think the professor would actually put something like that on a test anyway.

Thanks everyone!

But the real question is what are you going to do when something like this comes along in the real world? I maybe being a bit melodramatic, but he/she gave you the problem to address a serious disconnect between theory and practicality/reality, such that you may be better prepared to bridge the gap when you come upon it in the future.

Just a thought.

Originally posted by joe shmo
But the real question is what are you going to do when something like this comes along in the real world? I maybe being a bit melodramatic, but he/she gave you the problem to address a serious disconnect between theory and practicality/reality, such that you may be better prepared to bridge the gap when you come upon it in the future.

Just a thought.

I think the only human way to find the answer would be to plot it.

Since complex polynomial equations are not the point of the class, I don't feel like I need to spend a whole lot of time on it, especially since there is a lot of other material I could be studying and I have already spent a lot of time on this one problem.

I do see your point though. If I came across something like this in real life, I would probably contact my old statistics professor. 🙂

Statistics are like miniskirts.

What they reveal is tantalising, but what they hide is crucial.

Get your graphing calculator out.

Type the left side into y1. Type the right side into y2, then use the calculator's interset function to solve. TI command should be 2nd-trace-5-enter-enter-enter.

As long as the solution is on the screen, it will find it.

Originally posted by Eladar
Get your graphing calculator out.

Type the left side into y1. Type the right side into y2, then use the calculator's interset function to solve. TI command should be 2nd-trace-5-enter-enter-enter.

As long as the solution is on the screen, it will find it.

I have a TI-86 but don't have the manual, maybe I can find the directions online.

Originally posted by mlprior
I have a TI-86 but don't have the manual, maybe I can find the directions online.

Whilst you're playing around with graphing calculators and such; assuming you don't have money to burn grab yourself a copy of maxima 5.24 (poor man's mathematica but it's actually pretty good, and is actually the distant ancestor of the likes of maple and mathematica)
http://sourceforge.net/projects/maxima/files/

Also grab yourself a copy of Scilab (free clone of MATLAB - not quite as efficient but again, it's pretty good)
http://www.scilab.org/products/scilab/download

Then use these instead to do your plotting and root finding (and whatever else) - save the batteries on your TI :]


As for your OP, another pencil & paper approach might to perturb a trial solution x=1 with a small value a; i.e. evaluate 10(1+a)^9 -9(1+a)^10 = 3/4 and when you expand it you'll have a different degree 10 polynomial where you know already that the higher order terms are neglible (since |a| is small) and so you can throw some of these away - say up the the 4th power of a perhaps, and play around with that instead.

Originally posted by Agerg
Whilst you're playing around with graphing calculators and such; assuming you don't have money to burn grab yourself a copy of maxima 5.24 (poor man's mathematica but it's actually pretty good, and is actually the distant ancestor of the likes of maple and mathematica)
http://sourceforge.net/projects/maxima/files/

Also grab yourself a copy of Scilab (free ...[text shortened]... to do your plotting and root finding (and whatever else) - save the batteries on your TI :]

For equation solving or substitutes for graphic calculators, Wolfram Alpha is a pretty good site (see KN's post). No need to install software.