Question

Given a normal distribution with µ = 10 and σ = 2, find

(a) the normal curve area to the right of x = 6;
(b) the normal curve area between x = 6 and x = 14;
(c) the two values of x that contain the middle 75% of the normal curve area.

please show all work if possible. Thank you

Answer 1

Solution:

Given:

X follows a normal distribution with µ = 10 and σ = 2,

Part a) the normal curve area to the right of x = 6;

That is:

P( X > 6 )= .............?

Find z score for x = 6

Thus we get:

P( X > 6 )= P( Z> -2.00)

P( X > 6 )= 1 - P( Z < -2.00)

Look in z table for z = -2.0 and 0.00 and find corresponding area.

P( Z< -2.00 ) = 0.0228

thus

P( X > 6 )= 1 - P( Z < -2.00)

P( X > 6 )= 1 - 0.0228

P( X > 6 )= 0.9772

Part b) the normal curve area between x = 6 and x = 14;

P( 6 < X < 14) = .........?

Find z score for x = 14

Thus we get:

P( 6 < X < 14) = P( -2.00 < Z < 2.00 )

P( 6 < X < 14) = P( Z < 2.00 ) - P( Z< -2.00 )

Look in z table for z = 2.0 and 0.00 as well as for z = -2.0 and 0.00 and find corresponding area.

P( Z < 2.00 ) = 0.9772

thus

P( 6 < X < 14) = P( Z < 2.00 ) - P( Z< -2.00 )

P( 6 < X < 14) = 0.9772 - 0.0228

P( 6 < X < 14) = 0.9544

Part c) the two values of x that contain the middle 75% of the normal curve area.

P( x₁ < X < x₂ ) =0.75

Thus find z values such that:
P( -z < Z < z )= 0.75

Since middle area is 0.75, then area in tails = 1 - 0.75 = 0.25

Thus area in left tail = 0.25/2=0.125

Thus find z value such that:

P( Z < -z) =0.125

Look in z  table for Area = 0.1250 or its closest area and find corresponding z value.

Area 0.1251 is closest to 0.1250 and it corresponds to -1.1 and 0.05, that is -1.15

thus -z = -1.15

Since standard normal distribution is symmetric , P( Z< -z) =P( Z> z )

thus z = 1.15

thus we get two z values -1.15 and 1.15

Now use following formula to find x value:

and

thus the two values of x that contain the middle 75% of the normal curve area are: 7.7 and 12.3 .