Question

A random variable is normally distributed with a mean of 20 and a standard deviation of 4. If an observation is randomly selected from the distribution, a. What value will be exceeded 15% of the time? b. What value will be exceeded 75% of the time? c. Determine two values of which the smaller has 20% of the values below it and the larger has 20% of the values above it. d. What value will 25% of the observations be below?

Answer 1

a)

$$P(z>z')=0.15\to P(z<z')=0.85$$

From z-table:

$$z'=1.04$$

$$x'=20+(1.04)4=24.16$$

b)

$$P(z>z')=0.75\to P(z<z')=0.25$$

From z-table:

$$z'=-0.67$$

$$x'=20+(-0.67)4=17.32$$

c)

$$P(z<-z')=0.2, P(z>z')=0.2$$

From z-table:

$$P(z<-0.84)=0.2005$$

$$z'=0.84$$

$$x'_1=20+(-0.84)4=16.64$$

$$x'_2=20+(0.84)4=23.36$$

d)

$$P(z<z')=0.25$$

Thus, this is the same value as in the part b).

$$x'=17.32$$