Use the left graph of a normal distribution to answer the following 2 questions.

Question

Use the left graph of a normal distribution to answer the following 2 questions. Assume that the vertical lines are one standard deviation apart. What is the mean of this distribution? What is the standard deviation of this distribution? What is the formula that is used to transform normal distribution variable values, X, to standard -5 0 5 5 o 25 normal distribution variable values, z? What is the z-score at the most likely value of a standard normal distribution? What is the standard deviation of a standard normal distribution? Find P(-1.50

Explanation / Answer

Q1.
by observing graph, we can say that mean is always lies in the center value,
mean = 10
in graph, we see that 68% of data is lies b/2 mean-s.d, mean+s.d
68% of data is lies b/w ( 5,15)
and mean-s.d = 5, mean+s.d=15
=> with above we conlcude that s.d = 5

Q2.
the PDF of normal distribution is = 1/? * ?2? * e ^ -(x-u)^2/ 2?^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean ( u ) = 0
standard Deviation ( sd )= 1

Q3.
BETWEEN THEM
To find P(a < = Z < = b) = F(b) - F(a)
P(X < -1.5) = (-1.5-0)/1
= -1.5/1 = -1.5
= P ( Z <-1.5) From Standard Normal Table
= 0.0668
P(X < 0.78) = (0.78-0)/1
= 0.78/1 = 0.78
= P ( Z <0.78) From Standard Normal Table
= 0.7823
P(-1.5 < X < 0.78) = 0.7823-0.0668 = 0.7155

Q4.
LESS THAN OR GREATER THAN
To find P( X > a or X < b ) = P ( X > a ) + P( X < b)
P(X < -2.4) = (-2.4-0)/1
= -2.4/1= -2.4
= P ( Z <-2.4) From Standard Normal Table
= 0.0082
P(X > 2.3) = (2.3-0)/1
= 2.3/1 = 2.3
= P ( Z >2.3) From Standard Normal Table
= 0.0107
P( X < -2.4 OR X > 2.3) = 0.0082+0.0107 = 0.0189

Q5.
LESS THAN
P(X < 3.02) = (3.02-0)/1
= 3.02/1= 3.02
= P ( Z <3.02) From Standard Normal Table
= 0.9987

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