Question

Given a normal distribution with μ=40 and σ =9​, find​ (a) the normal curve area to the right of x=25; (b) the normal curve area to the left of x=29; (c) the normal curve area between x=43 and x=52​; (d) the value of x that has 90​% of the normal curve area to the​ left; and​ (e) the two values of x that contain the middle 70​% of the normal curve area.

Answer 1

Given a normal distribution with = 40 & = 9.

a) -

The normal curve area to the right of X = 25, i.e. P(X 25) -

P(X 25) = 1 - P(X < 25)

   ................

  

  

The normal curve area to the right of X = 25 is 0.9925.

b) -

The normal curve area to the left of X = 29, i.e. P(X 29) -

   ......................

  

The normal curve area to the left of X = 29 is 0.1112.

c) -

The normal curve area between X = 43 & X = 52 - P(43 X 52) -

.................

The normal curve area between X = 43 & X = 52 is 0.2418.

d) -

The value of X that has 90% of the normal curve area to the left -

...............

The value of X that has 90% of the normal curve area to the left is 51.61.

e) -

The two values of x that contains middle 70% of the normal curve -

Let, X & be the two values of X that contains middle 70% of the normal curve.

The curve is normal, so the area below x should be same as the area above .

Total area between x & is 0.7.

Hence, the remaining is 0.3.

So, the area below x is 0.15 & above is 0.15.

Hence,

..................

&

...................

Hence, The two values of x that contains middle 70% of the normal curve are 30.73 & 49.36.

Note : Z-tables provided below.