My professor and I have came to a disagreement. My problem is with question number 8. I am pretty sure that you have to use the Bayes'thm for the question. I have tried ONCE to convince him, but regrettably, I have failed. Here is the question(s). I have included the previous questions because some of the information (from previous questions) was used to solve the question 8.
For questions 5 and 6 use the following information: Diabetes - physicians recommend that children with type-1 diabetes keep up with insulin shots to minimize the chance of long-term complications. in addition, some diabetes researchers have observed that growth rate of weight during adolescence among diabetic patients is affected by level of compliance with insulin therapy. suppose 12 year old type-1 diabetic boys who comply with their insulin shots have a weight gain over 1 year that is normal distributed, with mean=12 lbs and variance =12 lbs.
Q5) what is the probability that compliant type-1 diabetic 12 year old boys will gain at least 15 pounds over 1 year? no continuity correction.
Q6) repeat problem #5 but now use the continuity correction (unit of measure is one pound).
Q7) Conversely, 12 year old type-1 diabetic boys who do not take their insulin shots (non compliant) have a weight gain over 1 year that is normally distributed with a mean of 8 pounds and a variance of 12 pounds. what is the probability that these non compliant boys will gain at least 15 pounds over 1 year? no continuity correction.
Q8) for the following problem, notice how the continuity correction is already built into the question: it is generally assumed that 75% of type-1 diabetics comply with their insulin regimen. suppose that a 12 year old type-1 diabetic boy comes to clinic and shows a 5lb weight gain over 1 year. (actually because of measurement error, assume this is an actual weight gain from 4.5 to 5.5 lbs). the boy claims to be taking his insulin medication. what is the probability that he is telling the truth?
I will jump right into the question. The question asked what is the P(he is telling the truth) i.e. pr(taking medication | weight gain of 5lbs). To me, this is unmistakably conditional.
find P(A|B) where A= taking medication B= weight gain of 5lbs.
Bayes' thm states... P(A|B) = P(B|A)P(A) / P(B)
Find P(B|A) i.e. what is the probability that he gains weight given he is taking med?
Given: mu=12 theta= root 12. Using the continuity correction...
area: min to 5.5 z = (x-mu)/theta = (5.5-12)/root 12 = -1.88 <-> A1 = 0.0301
area: min to 4.5 z = (x-mu)/theta = (4.5-12)/root 12 = -2.17 <-> A2 = 0.0150
Find P(A) i.e. what is the probability that he is taking med.
Given: In question 8, this was given as 75% or 0.75.
Find P(B) i.e. what is the probability that he gains weight of 5lbs?
Bayes' thm also states... P(B) = P(A int B)+(not A int B) = P(B|A)P(A) + P(B|not A)P(not A) i.e. P(weight gain given taking med) + P(weight gain given not taking med)
We already found P(weight gain given taking med) = 0.0151*
so find P(weight gain given not taking med)
Given:mu=8 theta=root 12. Using the continuity correction...
area: min to 5.5 z = (x-mu)/theta = (5.5-8)/root 12 = -0.72 <-> A1 = 0.2358
area: min to 4.5 z = (x-mu)/theta = (4.5-8)/root 12 = -1.01 <-> A2 = 0.1562
so, P(B) = 0.0151+0.0796
Putting P(B|A), P(A), and P(B) together...
P(A|B) = P(B|A)P(A) / P(B) = (0.0151)(0.75)/(0.0151+0.0796) = 0.1196 = 11.96%
The following is what my professor said about the question:
Jeff – As worded, it is not a conditional probability problem, therefore you do not use posterior probability. I strongly believe that the correct answer is the area from min to 5.5 (answer is then 0.0301). However most students found the area from 4.5 to 5.5 (answer is 0.015) because I mentioned the continuity correction and they were confused so I let this answer stand. In reality, from min to 5.5 would be non-compliance. In fact and specific number, even if compliant (e.g., Pr(X = 17)) would be a SMALL number. Thus the probability really should be 5 or less or Min to 5.5 with the continuity correction.
What I think he said was to find the area under the normal distribution curve (as in question 5) from min to 5.5, but that could not possibly be the final answer because the area under the curve is the probability that the boy gains 5lbs., assuming that he is compliant. I went to his office hour and asked him to explain himself. Unfortunately, I did not understand what he was saying.
I feel like I just took a crazy pill. I don't see any logical mistakes in my part. If there is any algebraic mistakes that I've made, please let me know. Thank you!
my best, Jeff Kwak