Question

The amounts of time per workout an athlete uses a stairclimber are normally​ distributed, with a mean of

22 minutes and a standard deviation of 7 minutes. Find the probability that a randomly selected athlete uses a stairclimber for​ (a) less than 17 ​minutes, (b) between 22 and 31

​minutes, and​ (c) more than  30 minutes.

​(a) The probability that a randomly selected athlete uses a stairclimber for less than 17 minutes is =

​(Round to four decimal places as​ needed.)

​(b) The probability that a randomly selected athlete uses a stairclimber between 22 and 31 minutes is =

​(Round to four decimal places as​ needed.)

​(c) The probability that a randomly selected athlete uses a stairclimber for more than 30 minutes is =

​(Round to four decimal places as​ needed.)

Answer 1

Part a)

X ~ N ( µ = 22 , σ = 7 )
P ( X < 17 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 17 - 22 ) / 7
Z = -0.7143
P ( ( X - µ ) / σ ) < ( 17 - 22 ) / 7 )
P ( X < 17 ) = P ( Z < -0.7143 )
P ( X < 17 ) = 0.2375

Part b)

X ~ N ( µ = 22 , σ = 7 )
P ( 22 < X < 31 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 22 - 22 ) / 7
Z = 0
Z = ( 31 - 22 ) / 7
Z = 1.2857
P ( 0 < Z < 1.29 )
P ( 22 < X < 31 ) = P ( Z < 1.29 ) - P ( Z < 0 )
P ( 22 < X < 31 ) = 0.9007 - 0.5
P ( 22 < X < 31 ) = 0.4007

Part c)

X ~ N ( µ = 22 , σ = 7 )
P ( X > 30 ) = 1 - P ( X < 30 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 30 - 22 ) / 7
Z = 1.1429
P ( ( X - µ ) / σ ) > ( 30 - 22 ) / 7 )
P ( Z > 1.1429 )
P ( X > 30 ) = 1 - P ( Z < 1.1429 )
P ( X > 30 ) = 1 - 0.8735
P ( X > 30 ) = 0.1265