Question

Find the following probability for the standard normal random variable z. a. ​P(zequals3​) e. ​P(minus3less than or equalszless than or equals3​) b. ​P(zless than or equals3​) f. ​P(minus1less than or equalszless than or equals1​) c. ​P(zless than3​) g. ​P(negative 2.66less than or equalszless than or equals0.06​) d. ​P(zgreater than3​) h. ​P(negative 0.75less thanzless than1.09​)

Answer 1

(a)

P(Z= 3) =0

EXPLANATION: For a continuous variable, probability at a particular point =0

(e)

P( - 3 Z 3) = 0.9973

EXPLANATION: By 68 - 95 - 99.7 Rule:

(b) P(Z3):

Table of Area Under Standard Normal Curve gives area = 0.4987

So,

P(Z3) = 0.5 + 0.4987 = 0.9987

(f)

P(-1Z<1) = 0.6827

EXPLANATION: By 68 - 95 - 99.7 Rule:

(c) P(Z<3) :

Table of Area Under Standard Normal Curve gives area = 0.4987

So,

P(Z<3) = 0.5 + 0.4987 = 0.9987

(g)
P(-2.66 Z0.06):

Case 1 :

For Z from - 2.66 to mid value:
Table gives area = 0.4971

Case 2; For Z from mid value to 0.06:

Table gives area = 0.0239

So,

P(-2.66 < Z< 0.06) =0.4971 + 0.0239 = 0.5210

(d)
P(Z>3)

Table gives area =0.4987

So,

P(Z>3) = 0.5 - 0.4987 = 0.0013

(h)

P(-0.75 < Z < 1.09):

Case1: For Z< - 0.75:

Table gives area =0.2734

Case2: For Z from mid value to 1.09:
Table gives area = 0.3621

So,

P(-0.75<Z<1.09) = 0.2734 + 0.3621 = 0.6355