Question 189250: A standard test for diabetes is based on glucose levels in the blood after fasting for a prescribed period. For untreated diabetics the mean fasting glucose is found to be 11.74mmol/L, and the standard deviation is 3.50mmol/L. The level of glucose appears to be approximately normally distributed.
To operate a simple diagnostic test based on fasting glucose levels, we need to set a cut off point, C, so that if the patients fasting glucose level is at least C we say they have diabetes. If it is lower we say they do not have diabetes.
(i) For the moment, suppose we use C=6.5. What is the probability that a diabetic is incorrectly diagnosed as not having diabetes?
NOTE: I have attempted this question, but got stuck, because I'm not sure if what I'm doing is correct. I found that the P(6.5 < x < 11.74) = 0.4325, but don't know where to go from here...
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! A standard test for diabetes is based on glucose levels in the blood after fasting for a prescribed period. For untreated diabetics the mean fasting glucose is found to be 11.74mmol/L, and the standard deviation is 3.50mmol/L. The level of glucose appears to be approximately normally distributed.
To operate a simple diagnostic test based on fasting glucose levels, we need to set a cut off point, C, so that if the patients fasting glucose level is at least C we say they have diabetes. If it is lower we say they do not have diabetes.
(i) For the moment, suppose we use C=6.5. What is the probability that a diabetic is incorrectly diagnosed as not having diabetes?
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Those with glucose above 6.5 will be diagnosed "diabetic"
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Those below 6.5 will be diagnosed "non-diabetic"
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Find the z-value of 6.5.
z(6.5) = (6.5-11.74)/3.5 = -1.4971
P(a person is diagnosed non-diabetic) = P(z < -1.4971) = 0.0672
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Cheers,
Stan H.
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