MHB -7.6.69 Determine the value z^* that...

  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Value
AI Thread Summary
The discussion focuses on determining specific z-values that separate different percentages of a standard normal distribution. For the largest 3%, the z-value is 1.88; for the largest 1%, it is 2.33; for the smallest 4%, it is 1.75; and for the smallest 10%, it is 1.28. The cumulative distribution function (CDF) of the standard normal distribution, denoted as Φ(z), is essential for these calculations. The inverse function, Φ^{-1}(p), is used to find the z-scores corresponding to given probabilities. Understanding these concepts is crucial for solving related statistical problems.
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
Determine the value $z^*$ that...
a. Separates the largest $3\%$ of all z values from the others
$=1.88$
b. Separates the largest $1\%$ of all z values from the others
$=2.33$
c. Separates the smallest $4\%$ of all z values from the others
$=1.75$
d. Separates the smallest $10\%$ of all z values from the others
$=1.28$

OK just can't seem to find an example of how these are stepped thru
the book answer follows the =
 
Mathematics news on Phys.org
once again $$\Phi(z)$$ is the CDF of the standard normal distribution
you'll need some way to compute the inverse of this function to complete this problem.

a)$$ \Phi(z^*) = 0.97\\ z^* = \Phi^{-1}(0.97)= 1.88079$$

b) $$\Phi(z^*) = 0.99$$

c) $$\Phi(z^*) = 0.04$$

d)$$ \Phi(z^*) = 0.1$$
 
romsek said:
once again $$\Phi(z)$$ is the CDF of the standard normal distribution
you'll need some way to compute the inverse of this function to complete this problem.

a)$$ \Phi(z^*) = 0.97\\ z^* = \Phi^{-1}(0.97)= 1.88079$$

b) $$\Phi(z^*) = 0.99$$

c) $$\Phi(z^*) = 0.04$$

d)$$ \Phi(z^*) = 0.1$$

mahalo I was unaware of the use of that symbol

Screenshot 2021-09-02 11.25.39 AM.png


ok I can see that at 1.88 goes to .97 on table
or using P to z calculator but still what is $\Phi^{-1}$
 
Last edited:
$$\Phi^{-1}(p)$$ is the inverse of $$\Phi(z)$$

If you are given a probability $$p, \Phi^{-1}(p)$$ returns the associated z-score of $$p$$
 
Since [math]\Phi(1.88)= 0.97[/math], [math]\Phi^{-1}(0.97)= 1.88[/math]
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top