Question

6. A normal distribution of has a mean of 20 and a standard deviation of 10. Find the z-scores corresponding to each of the following values: (10 points) a) What is the z score for a value of 30? b) What is the z score for a value of 10? c) What is the z score for a value of 15? d) What it P(20<x<30)? e) What is P (x > 10)? ) What is P (x < 15)? g) What is probability that z greater than 25 and less than 20 h) What z value corresponds to a probability of that x <31 )What z value corresponds to the 60h percentile? Ifz 2.5 what x value corresponds to this value.

Answer 1

a) The z score for a value of 30

30 - 20 10

b) The z score for a value of 10

10-20 10

c) The z score for a value of 15

15-20 10

d)

P(20 < X < 30) = P(X < 30) _ P(X < 20) = P(Z < 1)-P(Z <-1)

P(20< X < 30)- P(Z< 1)- P(Z<-1)

$\Rightarrow P(20 <X <30) = \Phi(1) - \Phi(-1)$

(20 < X < 30) = 0.841345-0.158655

P(20 < X < 30) 0.682690

e)

P(X > 10) = 1-P(X < 10) = 1-PlZ <-1) = 1-0.158655 = 0.841345

f)

P(X < 15) = P(Z <-0.5) 0.308538

g)

P(X > 25, X < 20)P(X>25) + P(X < 20)

X-20 10 25 - 20 10 X-20 10 20 - 20 10 P(X> 25, X < 20)>

P(X > 25, X < 20) = P(Z > 0.5)+P(Z < 0)

$\Rightarrow P(X>25, X<20) =1 - \Phi(0.5) + \Phi(0)$

P(X > 25. X < 20) = 0.308538+0.5

(X > 25, X < 20) = 0.808538

h) The z-score corresponds to a probability to that x < 31 = (31 - 20)/10= 1.1

i) The z-value corresponds to the 60th percentile = 0.253347

j) x value corresponds to z = 2.5 is = 20 + (2.5*10) = 20 + 12.5 = 32.5