If x is a normally distributed continuous random variable wi

If x is a normally distributed continuous random variable with mu = 25 and Sigma = 4, find the following: What is the probability that x is between 18 and 23.5? What is the probability that x is between 25 and 32? What percentage of the population will be greater than 20? What is the value of x where approximately 30% of the population lies less than x? What is the value of x where approximately 25% of the population lies greater than x? If x = 18.9, what is the percentile associated with x? Find the limits on x if x must fall in the middle 50% of all scores. Find the limits on x if only 5% of the population values are excluded from both the low values of x and the high value of x.

Solution

a).

z value for 18, z=(18-25)/4 =-1.75

z value for 23.5, z=(23.5-25)/4 = -0.38

p( 18<x<23.5) = P( -1.75<z<-0.38) =

=0.3520-0.0401

=0.3119

b).

z value for 25, z=(25-25)/4 =0

z value for 32, z=(32-25)/4 = 1.75

p( 25<x<32) = P( 0<z<1.75) =

=0.9599 – 0.5

=0.4599

c).

z value for 20, z=(20-25)/4 =-1.25

p( x >20) = p( z >-1.25) =0.8944

the required percentage =89.44%

d).

z value for lower 30% = -0.524

x= mean+z*sd

the required x=25-0.524*4 =22.904

e).

z value for top 25% = 0.674

the required x=25 +0.674*4 =27.696

f).

z value for 18.9, z=(18.9-25)/4 = -1.53

percentile associated with z= -1.53 = 6.3%

the required percentile =6.3%

g).

z values for middle 50% =(-0.674, 0.674)

the required lower x=25 -0.674*4 =22.304

the required upper x=25 +0.674*4 =27.696

h).

z values for middle 95% =(-1.96, 1.96)

the required lower x=25 -1.96*4 =17.16

the required upper x=25 +1.96*4 =32.84