Answer:
(a) 0.5899
(b) 0.9166
Step-by-step explanation:
Let X be the random variable that represents the height of a woman. Then, X is normally distributed with  
 = 62.5 in
 = 62.5 in
 = 2.2 in
 = 2.2 in
the normal probability density function is given by  
 , then
, then
(a)  = 0.5899
 = 0.5899
    (in the R statistical programming language) pnorm(63, mean = 62.5, sd = 2.2)
(b) We are seeking  where n = 37.
 where n = 37.  is normally distributed with mean 62.5 in and standard deviation
 is normally distributed with mean 62.5 in and standard deviation  . So, the probability density function is given by
. So, the probability density function is given by
 , and
, and
 = 0.9166
 = 0.9166
(in the R statistical programming language) pnorm(63, mean = 62.5, sd = 2.2/sqrt(37))
You can use a table from a book to find the probabilities or a programming language like the R statistical programming language.