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Answer to Question #22307 in Statistics and Probability for David

Question #22307
For a Normal distribution with mean, μ=2, and standard deviation, σ=4, 30% of all observations have a value less than

For a Normal distribution with mean, μ=2, and standard deviation, σ=4, 40% of all observations have a value greater than

For a normal distribution with mean, μ=2, and standard deviation, σ=4 what proportion of observations take values less than 1?

Can you also please explain how to get the answer?
Expert's answer
Let X be random variable having standard normaldistribution, that ismean μ=0, and
standard deviation, σ=1.

Let also F be the cumulative distribution function for X,so
F(t) = P(X<t)
The values of function F are computed and can be found inany book on probability theory.
So for problems 1)-3) we will use the values of functionF.
Let Y be the random variable having Normal distributionwith mean, μ=2, and standard deviation, σ=4.In 1) and 2) we should find values y1, y2 such that
1) P(Y < y1) = 0.32) P(Y > y2) = 0.4
In 3) we should find probability
3) P(Y < 1).
Consider the so called "normalized" randomvariable Z = (Y-μ)/σ =(Y-2)/4.
Then Z has standard normal distribution and so
P(Z<t) = F(t).
Now we can solve 1)-3).1)
0.3 = P(Y<y1) = P((Y-2)/4 < (y1-2)/4 )
= P( Z <(y1-2)/4 )
= F( (y1-2)/4)
From tables for F we obtain that F(-0.524) = 0.3
whence
(y1-2)/4 = -0.542
Therefore
y1 = -0.542*4+2=-0.168
Thus for a Normal distribution with mean, μ=2, andstandard deviation, σ=4,
30% of all observations have a value less than -0.1682)

0.4 = P(Y>y2)
= P((Y-2)/4 > (y1-2)/4 )
= P( Z >(y2-2)/4 )
= 1 - P( Z< (y2-2)/4 )
= 1 - F((y2-2)/4 )
whence F( (y2-2)/4 ) =1-0.4=0.6.

From tables for Fwe obtain that
F(0.253) = 0.6
whence
(y2-2)/4 = 0.253
Therefore
y1 = 0.253*4+2=3.012

Thus for a Normal distribution with mean, μ=2, andstandard deviation, σ=4,
40% of all observations have a value greater than 3.0123)

P(Y < 1) = P((Y-2)/4 < (1-2)/4 )
= P( Z< -1/4 )
= P( Z< -0.25 )
=F(-0.25)
= 0.401

Thus for a normal distribution with mean, μ=2, andstandard deviation, σ=4
40.1 % of observations take values less than 1

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