Question

For 300 trading​ days, the daily closing price of a stock​ (in $) is well modeled by a Normal model with mean ​$196.55 and standard deviation ​$7.17. According to this​ model, what cutoff value of price would separate the ​

a) lowest 17​% of the​ days? ​

b) highest 0.78​%? ​

c) middle 61​%? ​

d) highest 50​%?

Answer 1

Part a)

P ( X < ? ) = 17% = 0.17
Looking for the probability 0.17 in standard normal table to calculate critical value Z = -0.95

-0.95 = ( X - 196.55 ) / 7.17
X = 189.7385
P ( X < 189.7385 ) = 0.17

Part b)

P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.0078 = 0.9922
Looking for the probability 0.9922 in standard normal table to calculate critical value Z = 2.42

2.42 = ( X - 196.55 ) / 7.17
X = 213.9014
P ( X > 213.9014 ) = 0.0078

part c)

P ( a < X < b ) = 0.61
Dividing the area 0.61 in two parts we get 0.61/2 = 0.305
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.305
Area above the mean is b = 0.5 + 0.305
Looking for the probability 0.195 in standard normal table to calculate critical value Z = -0.86
Looking for the probability 0.805 in standard normal table to calculate critical value Z = 0.86

-0.86 = ( X - 196.55 ) / 7.17
a = 190.3838
0.86 = ( X - 196.55 ) / 7.17
b = 202.7162
P ( 190.3838 < X < 202.7162 ) = 0.61

Part d)

P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.5 = 0.5
Looking for the probability 0.5 in standard normal table to calculate critical value Z = 0

0 = ( X - 196.55 ) / 7.17
X = 196.55
P ( X > 196.55 ) = 0.5