Question

# For 300 trading days, the daily closing price of a stock (in \$) is well modeled...

For 300 trading days, the daily closing price of a stock (in \$) is well modeled by a Normal model with mean \$195.61 and standard deviation \$7.15. According to this model, what cutoff value of price would separate the

a) lowest 11% of the days?

b) highest 0.86%?

c) middle 58%?

d) highest 50%?

Part a) P ( X < ? ) = 11% = 0.11
Looking for the probability 0.11 in standard normal table to calculate critical value Z = -1.23 -1.23 = ( X - 195.61 ) / 7.15
X = 186.82
P ( X < 186.82 ) = 0.11

Part b) P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.0086 = 0.9914
Looking for the probability 0.9914 in standard normal table to calculate critical value Z = 2.38 2.38 = ( X - 195.61 ) / 7.15
X = 212.63
P ( X > 212.63 ) = 0.0086

Part c) P ( a < X < b ) = 0.58
Dividing the area 0.58 in two parts we get 0.58/2 = 0.29
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.29
Area above the mean is b = 0.5 + 0.29
Looking for the probability 0.21 in standard normal table to calculate critical value Z = -0.81
Looking for the probability 0.79 in standard normal table to calculate critical value Z = 0.81 -0.81 = ( X - 195.61 ) / 7.15
a = 189.82
0.81 = ( X - 195.61 ) / 7.15
b = 201.40
P ( 189.82 < X < 201.40 ) = 0.58

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 - 195.61 ) / 7.15
X = 195.61
P ( X > 195.61 ) = 0.5

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