Question

For 300 trading​ days, the daily closing price of a stock​ (in $) is well modeled by a Normal model with mean $195.89 and standard deviation ​$7.18

According to this​ model, what cutoff value of price would separate the

​a) lowest 15% of the​ days?

​b) highest 0.61​%?

​c) middle 80​%?

​d) highest 50​%?

c) Select the correct answer below and fill in the answer​ box(es) within your choice.

A.The cutoff points are _________ and _________.

​(Use ascending order. Round to two decimal places as​ needed.)

B.The cutoff point is _________. ​(Round to two decimal places as​ needed.)

Answer 1

Part a)

X ~ N ( µ = 195.89 , σ = 7.18 )
P ( X < ? ) = 15% = 0.15
Looking for the probability 0.15 in standard normal table to calculate critical value Z = -1.04
Z = ( X - µ ) / σ
-1.04 = ( X - 195.89 ) / 7.18
X = 188.4228   188.42
P ( X < 188.4228 ) = 0.15

The cutoff points is  X = 188.4228   188.42

Part b)

X ~ N ( µ = 195.89 , σ = 7.18 )

0.61 % = 0.61/100 = 0.0061
P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.0061 = 0.9939
Looking for the probability 0.9939 in standard normal table to calculate critical value Z = 2.51
Z = ( X - µ ) / σ
2.51 = ( X - 195.89 ) / 7.18
X = 213.9118
P ( X > 213.9118 ) = 0.0061

The cutoff points is  X = 213.9118 213.91

Part c)

X ~ N ( µ = 195.89 , σ = 7.18 )
P ( a < X < b ) = 0.8
Dividing the area 0.8 in two parts we get 0.8/2 = 0.4
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.4
Area above the mean is b = 0.5 + 0.4
Looking for the probability 0.1 in standard normal table to calculate critical value Z = -1.28
Looking for the probability 0.9 in standard normal table to calculate critical value Z = 1.28
Z = ( X - µ ) / σ
-1.28 = ( X - 195.89 ) / 7.18
a = 186.6996
1.28 = ( X - 195.89 ) / 7.18
b = 205.0804
P ( 186.6996 < X < 205.0804 ) = 0.8

The cutoff points are

a = 186.6996   186.70

b = 205.0804   205.08

Part d)

X ~ N ( µ = 195.89 , σ = 7.18 )
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
Z = ( X - µ ) / σ
0 = ( X - 195.89 ) / 7.18
X = 195.89
P ( X > 195.89 ) = 0.5
The cutoff points is X = 195.89