Assume that X is normal with mean 10 and variance 25 Please

Assume that X is normal with mean 10 and variance 25.

Please show step by step

Assume that X is normal with mean 10 and variance 25. Find (a) P(X

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    9      
u = mean =    10      
          
s = standard deviation = sqrt(25) =     5      
          
Thus,          
          
z = (x - u) / s =    -0.2      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -0.2   ) =    0.420740291 [ANSWER]

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

b)

We first get the z score for the critical value. As z = (x - u) / 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 right tailed area of this is          
          
P(z >   0.2   ) =    0.420740291 [ANSWER]

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

c)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    9      
x2 = upper bound =    11      
u = mean =    10      
          
s = standard deviation =    5      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.2      
z2 = upper z score = (x2 - u) / s =    0.2      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.420740291      
P(z < z2) =    0.579259709      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.158519419 [ANSWER]