Answer:
a) 90th percentile
b) X = 51.43 ( Sophia's height ) , Amy's height = 50.92
c) 0.2394
d) 46.44 inches
Step-by-step explanation:
mean value( <em>u</em> ) = 47.7 inches
std ( σ )= 2.4 inches
Amy height = 1.34 std above mean value
Sophia height = 94th percentile of height distribution
A ) what percentile is Amy's height
To Calculate the percentile of Amy's height
P( X ≤ 50.916 ) = P( (X-µ)/σ ) ≤ (50.916-47.7) /2.4)
=P(Z ≤ 1.340 ) = 0.9099 = 90th percentile
B) How Amy's height compare to Sophia's
The proportion given based on Sophia's height = 0.94
z -value at the proportion ( 0.94 ) = 1.55 ( from excel )
note z-value can be calculated using = ( x - µ ) / σ
therefore: X (Sophia's height ) = zσ + µ= (1.55 * 2.4) + 47.7
X = 51.43 ( Sophia's height ) , Amy's height = 50.92
C) percentage expected to be taller than 49.4 inches
P ( X ≥ 49.40 ) = P( (X-µ)/σ) ≥ (49.4-47.7) / 2.4)
= P(Z ≥ 0.708 )
= P( Z < -0.708 )
= 0.2394
D) Below what height are the shortest 30% of 7-year old girls
( <em>u</em> ) = 47.7 inches
std ( σ )= 2.4 inches
given proportion = 30% = 0.3
The z value at 0.3 = -0.52 ( from excel )
to calculate for X ( height ) we have to apply the formula used to determine z value
z= (x-µ) /σ
hence : X = (Z * σ) +<em> u = </em>(-0.52 * 2.4 ) + 47.7
X = 46.44
below 46.44 inches lies about 30% of 7 year old girls height
and 
which is further split into