8 In a diabetes study fasting blood glucose levels were meas

8. In a diabetes study, fasting blood glucose levels were measured for a large number of participants. The population was then followed for 10 years. At the end of this time, it is found among individuals who eventually developed diabetes, their mean fasting blood glucose level was normally distributed with mean 140 mg/dL and variance 400 mg/dL. For those who did not develop diabetes, their mean fasting blood glucose level was normally distributed with mean 110 mg/dL and variance 256 mg/dL. (
a) What is the probability that a randomly selected person from those who eventually developed diabetes has a fasting blood glucose level greater than 125 mg/dL?
b) If we use fasting blood glucose level as a diagnostic test for diabetes, what cutoff value shall we use in order to have a sensitivity of 90%?
c) If we use the cutoff value that makes the test have a sensitivity of 90%, what is the specificity of this test?
d) If a sample of size 16 is selected from the healthy participants (i.e., those who did not develop diabetes), what’s the probability that this sample will have a mean fasting blood glucose level between 110 and 115 mg/dL?
e) What is the probability that 3 of the 16 healthy participants selected have a fasting blood glucose level between 110 and 115 mg/dL
f) Assume the fasting blood glucose levels of those who eventually developed diabetes and the healthy participants (i.e., those who did not develop diabetes) are independent of each other. If we randomly select a participant who developed diabetes and another participant who did not develop diabetes, what is the probability that both of them had a fasting blood glucose level greater than 125 mg/dL?
g) If repeated samples are selected from the healthy population, how large must the samples be for 95% of the sample means to lie within 1 mg/dL of the population mean?

Solution

X - For diabetes developed persons Blood glucose level is normal (140, 20 )

Y - For those who did not develop diabetes, fasting blood glucose level was normally distributed (110,16)

a) P(X>125) = P(Z>-0.75_

= 0.7734

b) 90th percentile is 1.28

X score = 140+1.28(20)

= 165.60