# Statistics Homework

mlprior
Science 23 Apr '11 23:03
1. 23 Apr '11 23:03
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
2. Palynka
Upward Spiral
24 Apr '11 03:53
The equation does not seem invertible in x so I don't think you can solve it analytically.
3. AThousandYoung
24 Apr '11 04:53
What if you repeatedly take the derivative with respect to x until you have a quadratic? Will that work?
4. Palynka
Upward Spiral
24 Apr '11 10:29
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?
5. 24 Apr '11 15:40
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
6. joe shmo
Strange Egg
24 Apr '11 15:55
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))
7. AThousandYoung
24 Apr '11 17:17
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.
8. 24 Apr '11 21:55
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!
9. joe shmo
Strange Egg
25 Apr '11 00:45
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.
10. 25 Apr '11 03:53
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. ðŸ™‚
11. Palynka
Upward Spiral
25 Apr '11 13:02
Statistics are like miniskirts.

What they reveal is tantalising, but what they hide is crucial.
12. 25 Apr '11 13:54

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.
13. 25 Apr '11 17:09

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.
14. Agerg
The 'edit'or
25 Apr '11 17:214 edits
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)

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.
15. Palynka
Upward Spiral
25 Apr '11 17:35
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.