Consider a random variable Z that follows the standard Norma

Consider a random variable Z, that follows the standard Normal distribution with mean = 0 and standard deviation = 1.

a) What is the probability that an outcome z is greater than 2.60?

b) What is the probability that z is less than 1.35?

c) What is the probability that z is between -1.70 and 3.10?

d) What value of z cuts off the upper 15% of the standard normal distribution?

Solution

a)

Using a table/technology, the right tailed area of this is          
          
P(z >   2.6   ) =    0.004661188 [answer]

b)

Using a table/technology, the left tailed area of this is          
          
P(z <   1.35   ) =    0.911492009 [answer]

c)

z1 = lower z score =    -1.7      
z2 = upper z score =     3.1      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.044565463      
P(z < z2) =    0.999032397      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.954466934   [answer]

d)

The right tailed area is 0.15, so the left tailed area is 1-0.15 = 0.85.

We get the z score from the given left tailed area. As          
          
Left tailed area =    0.85      
          
Then, using table or technology,          
          
z =    1.036433389   [answer]