A variable is normally distributed with mean 18 and standard deviation 3. a. Det

Question

A variable is normally distributed with mean

18

and standard deviation

3.

a. Determine the quartiles of the variable.

b. Obtain and interpret the

85th

percentile.

c. Find the value that 65% of all possible values of the variable exceed.

d.

Find the two values that divide the area under the corresponding normal curve into a

middle area of 0.95 and two outside areas of 0.025. Interpret the answer.

d.

Find the two values that divide the area under the corresponding normal curve into a

middle area of 0.95 and two outside areas of 0.025. Interpret the answer.

Explanation / Answer

Mean ( u ) =18
Standard Deviation ( sd )=3
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a.
Q1 = P ( Z < x ) = 0.25
Value of z to the cumulative probability of 0.25 from normal table is -0.674
P( x-u/s.d < x - 18/3 ) = 0.25
That is, ( x - 18/3 ) = -0.67
--> x = -0.67 * 3 + 18 = 15.9765                  
Q2 = P ( Z < x ) = 0.5
Value of z to the cumulative probability of 0.5 from normal table is 0
P( x-u/s.d < x - 18/3 ) = 0.5
That is, ( x - 18/3 ) = 0
--> x = 0 * 3 + 18 = 18                  
Q3 = P ( Z < x ) = 0.75
Value of z to the cumulative probability of 0.75 from normal table is 0.674
P( x-u/s.d < x - 18/3 ) = 0.75
That is, ( x - 18/3 ) = 0.67
--> x = 0.67 * 3 + 18 = 20.0235                  
b.
P ( Z < x ) = 0.85
Value of z to the cumulative probability of 0.85 from normal table is 1.036
P( x-u/s.d < x - 18/3 ) = 0.85
That is, ( x - 18/3 ) = 1.04
--> x = 1.04 * 3 + 18 = 21.1093                  
c.
P ( Z > x ) = 0.65
Value of z to the cumulative probability of 0.65 from normal table is -0.3853
P( x-u/ (s.d) > x - 18/3) = 0.65
That is, ( x - 18/3) = -0.3853
--> x = -0.3853 * 3+18 = 16.844                  
d.
P ( Z < x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is -1.96
P( x-u/s.d < x - 18/3 ) = 0.025
That is, ( x - 18/3 ) = -1.96
--> x = -1.96 * 3 + 18 = 12.1201                  
P ( Z > x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is 1.96
P( x-u/ (s.d) > x - 18/3) = 0.025
That is, ( x - 18/3) = 1.96
--> x = 1.96 * 3+18 = 23.8799                  
under the corresponding normal curve into a middle area of 0.95 are [ 12.1201, 23.8799   ]

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