Question

In a survey of U.S. women, the heights in the heights in the 20-29 age group were normally distributed, with a mean of 64.2 inches and standard deviation of 2.9 inches. Find the probability that a randomly selected study participant has a height that is (a) less than 56.5 inches, (B) between 61 and 67 inches, and (c) more than 70.5 inches, and (D) indentify any unusual events, Explain your reasoning

Answer 1

Part a)

P ( X < 56.5 )
Standardizing the value

Z = ( 56.5 - 64.2 ) / 2.9
Z = -2.66

P ( X < 56.5 ) = P ( Z < -2.66 )
P ( X < 56.5 ) = 0.0039

Part b)

P ( 61 < X < 67 )
Standardizing the value

Z = ( 61 - 64.2 ) / 2.9
Z = -1.1
Z = ( 67 - 64.2 ) / 2.9
Z = 0.97
P ( -1.1 < Z < 0.97 )
P ( 61 < X < 67 ) = P ( Z < 0.97 ) - P ( Z < -1.1 )
P ( 61 < X < 67 ) = 0.8329 - 0.1349
P ( 61 < X < 67 ) = 0.6979

Part c)

P ( X > 70.5 ) = 1 - P ( X < 70.5 )
Standardizing the value

Z = ( 70.5 - 64.2 ) / 2.9
Z = 2.17

P ( Z > 2.17 )
P ( X > 70.5 ) = 1 - P ( Z < 2.17 )
P ( X > 70.5 ) = 1 - 0.985
P ( X > 70.5 ) = 0.015

Part d)

indentify any unusual events

Events in part a and c are unusual, because the probability is less than 5% i.e  

Probability is less than 0.05.