Answer
a) $P( x < 200) = 0.2177$
b) $P( 250 < $x$ < 350)$ = 0.4133
c) $P( x> 500) = 0.0034$
Work Step by Step
PART A
To find P( $x$ < 200):
Step 1: Convert $x$ = 200 to a z score:
$z = \frac{200 - 267}{86} = -0.78$
Step 2: Find $P( z < -0.78)$
= $0.2177$
Therefore, $P( x < 200) = 0.2177$
PART B
To find $P( 250 < $x$ < 350)$:
Step 1: Convert $x = 250$ to a z score:
z = $\frac{250 - 267}{86} = -0.20$
Step 2: Convert $x = 350$ to a z score:
z = $\frac{350 - 267}{86} = 0.97$
Step 3: P( -0.20 < z < 0.97)
=P(z < 0.97) - P(z < -0.20)
= 0.8340 - 0.4207
= 0.4133
Therefore, $P( 250 < $x$ < 350)$ = 0.4133
PART C
To Find $P( x > 500)$
Step 1: Convert 500 into a z score
z = $\frac{500 - 267}{86} = 2.71$
Step 2: Find $P( z > 2.71)$
= $ 1- P( z < 2.71)$
= $ 1 - 0.9966$
= $0.0034$
