Question

For 300 trading​ days, the daily closing price of a stock​ (in $) is well modeled by a Normal model with mean ​$196.59

and standard deviation $7.16.

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

​a) lowest 14​% of the​ days?

​b) highest 0.42​%?

​c) middle 63​%?

​d) highest 50​%?

Answer 1

Solution :

mean = = 196.59

standard deviation = = 7.16

Using standard normal table,

(a)

P(Z < z) = 14%

P(Z < -1.08) = 0.14

z = -1.08

Using z-score formula,

x = z * +

x = -1.08 * 7.16 + 196.59 = 188.86

Cuttoff = 188.86

(b)

P(Z > z) = 0.42%

1 - P(Z <z) = 0.0042

P(Z < z) = 1 - 0.0042 = 0.9958

P(Z < 2.64) = 0.9958

z = 2.64

Using z-score formula,

x = z * +

x = 2.64 * 7.16 + 196.59 = 215.49

Cuttoff = 215.49

c)

1 - 63% = 37%

37% / 2 = 16.5%

P(Z < z) = 0.165

P(Z < -0.97) = 0.165

z = -0.97

Using z-score formula,

x = z * +

x = -0.97 * 7.16 + 196.59 = 189.72

P(Z > z) = 0.165

1 - P(Z < z) = 0.165

P(Z < z) = 1 - 0.165 = 0.835

P(Z < 0.97) = 0.835

z = 0.97

Using z-score formula,

x = z * +

x = 0.97 * 7.16 + 196.59 = 203.46

Cutoff = 189.72 and 203.46

d)

P(Z > z) = 50%

1 - P(Z < z) = 0.50

P(Z < z) = 1 - 0.50 = 0.50

P(Z < 0) = 0.50

z = 0

Using z-score formula,

x = z * +

x = 0 * 7.16 + 196.59 = 196.59

Cutoff = 196.59