Question

Please give a step by step solution:

The ages of a group of 50 women are approximately normally distributed with a mean of 50 years and a standard deviation of 55 years. One woman is randomly selected from the​ group, and her age is observed.

a. Find the probability that her age will fall between 56 and 59years.

b. Find the probability that her age will fall between 4747 and 51 years.

c. Find the probability that her age will be less than 35 years.

d. Find the probability that her age will exceed 41 years.

Answer 1

Solution :

Given that,

mean = = 50

standard deviation = = 55

n = 1

= 50

= / n = 55 / 1 = 55

( a) P(56< < 59)  

= P[(56 - 50) / 55< ( - ) / < (59 - 50) /55 )]

= P( 0.11< Z < 0.16)

= P(Z < 0.16) - P(Z <0.11 )

Using z table,  

= 0.5636 - 0.5438

probability = 0.0198

(b)

P(47< < 51)  

= P[(47 - 50) / 55< ( - ) / < (51 - 50) /55 )]

= P( -0.05< Z < 0.02)

= P(Z < 0.02) - P(Z < -0.05)

Using z table,  

= 0.508 - 0.4801

probability = 0.0279

(c)

P( < 35) = P(( - ) / < (35 - 50) /55 )

= P(z < -0.27)

Using z table

probability= 0.3936

(d)

P( >41 ) = 1 - P( < 41)

= 1 - P[( - ) / < (41 - 50) /55 ]

= 1 - P(z <-0.16 )

Using z table,    

= 1 - 0.4364

probability =0.5636