What relationship exists between the standard normal distribution and the box-plot methodology for describing distributions of data by means of quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal. Calculate the z values that correspond to the outer fences of the box plot for a normal probability distribution.

Slovenujozk 2022-09-12 Answered
What relationship exists between the standard normal distribution and the box-plot methodology for describing distributions of data by means of quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal. Calculate the z values that correspond to the outer fences of the box plot for a normal probability distribution.
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Answers (1)

Zayden Dorsey
Answered 2022-09-13 Author has 18 answers
In this exercise we need to show what kind of relationship there is between the standard normal distribution and the box-plot methodology for describing distributions of data by means of quartiles.
In this part we need to find the lower outer fence of the box-plot and upper inner outer of the box-plot for the standard normal random variable Z.
We know that the lower outer fence of a box-plot is equal to:
Q L 3 I Q R ,
and the upper outer fence of a box-plot is equal to:
Q U + 3 I Q R .
We also know that the interquartile range (IQR) is equal to:
I Q R = Q U Q L
From part a) we have:
Q L = z L = .675 ,
Q U = z U = .675
Then we have:
IQR=.675-(-.675)=.675+.675=1.35.
According to this the lower outer fence of the box-plot and upper outer fence of the box-plot for the standard normal random variable Z is:
Lower outer fence =-.675-3*1.35=-.675-4.05=-4.725,
Upper outer fence =.675+3*1.35=.675+4.05=4.725.
Result:
Lower outer fence =-4.725; Upper outer fence =4.725

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New questions

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
x ( k + 1 ) = A x ( k ) + b u ( k ) , y ( k ) = c x ( k )
where b = ( 0 , 1 ) T , c = ( 1 , 0 ) , A = [ 2 1 0 g ] for some g R
Find a feedback regulation (if there is any) of the form u ( k ) = K x ^ ( k ) where x ^ ( k ) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e ( k ) = x ( k ) x ^ ( k ) go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are λ 1 = 2 , λ 2 = g (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle | z | = 1.
The state observer equation is
[ x ( k + 1 ) e ( k + 1 ) ] T = [ A b K B k O A L C ] [ x ( k ) e ( k ) ] T
With characteristic equation
χ ( z ) = | z I A + b K | | z I A + L C | = χ K ( z ) χ L ( z )
Also consider
K = [ k 1 k 2 k 3 k 4 ]
and let a = k 1 + k 3 , β = k 2 + k 4
Then χ K ( z ) = ( z 2 ) ( z + g + β ) + a.
So we can select some eigenvalues inside the unit circle and determine a , β in terms of g. Choosing e.g. λ 1 , 2 = ± 1 / 2 we get a = 3 g + 33 / 8 , β = 9 / 4 g , g R
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose | λ 1 , 2 | < 1?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)