Question

Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 miligrams per deciliter (mg/dl) one hour after having a sugary drink. Sheila's measured glucose level one hour after the sugary drink varies according to the Normal distribution with μμ = 125 mg/dl and σσ = 15 mg/dl.

(a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes?
(b) If measurements are made on 6 separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes?

Answer 1

a)

X ~ N ( µ = 125 , σ = 15 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 140 ) = P(Z > (140 - 125 ) / 15 )
= P ( Z > 1 )
= 1 - P ( Z < 1 )
= 1 - 0.8413
= 0.1587

b)

X ~ N ( µ = 125 , σ = 15 )  
P ( X > 140 ) = 1 - P ( X < 140 )   
Standardizing the value   
Z = ( X - µ ) / ( σ / √(n))  
Z = ( 140 - 125 ) / ( 15 / √ ( 6 ) )  
Z = 2.45  
P ( ( X - µ ) / ( σ / √ (n)) > ( 140 - 125 ) / ( 15 / √(6) )  
P ( Z > 2.45 )   
P ( X̅ > 140 ) = 1 - P ( Z < 2.45 )   
= 1 - 0.9929  
= 0.0071