Question

The score on an test from a certain statistics class, ?, is normally distributed with ?=80.8 and ?=8.3.

NOTE: Assume for the purpose of this problem that the score is a continuous variable. A score can thus take on any value on the continuum. (In real life, scores are often treated as if they were continuous values but are actually discrete in most cases.)

(a) Write the event ''a score over 65.8'' in terms of ?:  .

(b) Find the probability of this event:

(c) Find the probability that a randomly chosen score is greater than 87.8:  .

(d) Find the probability that a randomly chosen score is between 65.8 and 87.8:

Answer 1

a)

A score over 65.8 = X > 65.8

b)

P ( X > 65.8 ) = 1 - P ( X < 65.8 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 65.8 - 80.8 ) / 8.3
Z = -1.81
P ( ( X - µ ) / σ ) > ( 65.8 - 80.8 ) / 8.3 )
P ( Z > -1.81 )
P ( X > 65.8 ) = 1 - P ( Z < -1.81 )
P ( X > 65.8 ) = 1 - 0.0351
P ( X > 65.8 ) = 0.9649

c)

P ( X > 87.8 ) = 1 - P ( X < 87.8 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 87.8 - 80.8 ) / 8.3
Z = 0.84
P ( ( X - µ ) / σ ) > ( 87.8 - 80.8 ) / 8.3 )
P ( Z > 0.84 )
P ( X > 87.8 ) = 1 - P ( Z < 0.84 )
P ( X > 87.8 ) = 1 - 0.7995
P ( X > 87.8 ) = 0.2005

d)

P ( 65.8 < X < 87.8 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 65.8 - 80.8 ) / 8.3
Z = -1.81
Z = ( 87.8 - 80.8 ) / 8.3
Z = 0.84
P ( -1.81 < Z < 0.84 )
P ( 65.8 < X < 87.8 ) = P ( Z < 0.84 ) - P ( Z < -1.81 )
P ( 65.8 < X < 87.8 ) = 0.7995 - 0.0351
P ( 65.8 < X < 87.8 ) = 0.7644