Data from the article The Osteological Paradox Problems infe

Data from the article \"The Osteological Paradox: Problems inferring Prehistoric Health from Skeletal Samples\" (Current Anthropology (1992):343-370) suggests that a reasonable model for the distribution of heights of 5-year old children (in centimeters) is N(100, 62) . Let the letter X represent the variable \"height of 5-year old\", and use this information to answer the following. Use 4 decimal places unless otherwise indicated.

(a) P(X > 89.2) =  

(b) P(X < 109.78) =  

(c) P(97 < X < 106) =  

(d) P(X < 85.6 or X > 111.4) =  

(e) P(X > 103) =  

(f) P(X < 98.2) =  

(g) P(100 < X < 124)=  

(h) The middle 80% of all heights of 5 year old children fall between __ and __ . (Use 2 decimal places.)

Solution

a)

P(X>89.2) = P(Z > 89.2 - 100/62) = P(Z > -0.1742) = 0.5691

b)

P(X<109.78) = P(Z < 109.78 - 100 /62) = 0.5627

c)
P(97 < X< 106) = P(Z < 106 - 100/62) - P(Z < 97 - 100 /62)

= 0.5385 - 0.5193

= 0.0192

d)

P(X<85.6 or X > 114) = P(X < 85.6) + P(X > 114)

= P( Z < 85.6 -100/62) + P( Z > 114 -100/62)

= 0.4082 + 0.4107

= 0.8189

e)
P(X > 103) = P(Z > 103 -100/62) = 0.4807