g The distribution of heights of 7-year-old girls is approximately normally distributed with a mean of 47.7 inches and a standard deviation of 2.4 inches. Amy is 7 years old, and her height is 1.34 standard deviations above the mean. Her friend Sophia is also 7 years old and is at the 94th percentile of the height distribution. a. At what percentile is Amy's height? Draw and label the Normal model that corresponds to this situation. b. How does her height compare to Sophia's? c. What percent of 7-year-old girls would you expect to be taller than 49.4 inches? Draw and label the Normal model that corresponds to this situation. d. Below what height are the shortest 30% of 7-year-old girls? Draw and label the Normal model that corresponds to this situation.

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( u ) = 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

( u ) = 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)  + u = (-0.52 * 2.4 ) + 47.7

            X = 46.44

below 46.44 inches lies about 30% of 7 year old girls height

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the answer is 40.

proof: