Shelia'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 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 129 mg/dl and standard deviation 8 mg/dl. Let L denote a patient's glucose level.
(a) If measurements are made on four different days, find the level L such that there is probability only 0.01 that the mean glucose level of four test results falls above L for Shelia's glucose level distribution. What is the value of L?
(b) If the mean result from the four tests is compared to the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes?
a) P( > L) = 0.01
or, P(( - )/() > (L - )/()) = 0.01
or, P(Z > (L - 129)/(8/)) = 0.01
or, P(Z < (L - 129)/4) = 0.99
or, (L - 129)/4 = 2.33
or, L = 2.33 * 4 + 129
or, L = 138.32
b) P( > 140)
= P(( - )/() > (140 - )/())
= P(Z > (140 - 129)/(8/))
= P(Z > 2.75)
= 1 - P(Z < 2.75)
= 1 - 0.9970
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