Let X N10 5 2 Answer the following questions about X a Find

Let X N(10, 5 2 ). Answer the following questions about X.

(a) Find P(X 11).

(b) Find P(X 8.4).

(c) Find x such that P(X x) = 0.95.

Solution

Do you mean N(10, 5^2)?

a)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    11      
u = mean =    10      
          
s = standard deviation =    5      
          
Thus,          
          
z = (x - u) / s =    0.2      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   0.2   ) =    0.579259709 [answer]

**************

b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    8.4      
u = mean =    10      
          
s = standard deviation =    5      
          
Thus,          
          
z = (x - u) / s =    -0.32      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.32   ) =    0.625515835 [answer]

****************

c)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.95      
          
Then, using table or technology,          
          
z =    1.644853627      
          
As x = u + z * s / sqrt(n)          
          
where          
          
u = mean =    10      
z = the critical z score =    1.644853627      
s = standard deviation =    5      
          
Then          
          
x = critical value =    18.22426813   [answer]