# What is the probability that the boy is telling the truth?

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.

start

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?

end

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

A1-A2=0.0151

P(B|A)=0.0151*

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

A1-A2=0.0796

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:

start

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.

end

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

-
What is min? –  PEV Jan 16 '11 at 4:27
Sorry. Min = minimum. –  Jeff Kwak Jan 16 '11 at 4:28
minimum of what? –  PEV Jan 16 '11 at 4:29
The normal distribution curve is a bell shaped curve. We are finding area under the curve. Minimum in this case would be 0. Although the patients could lose weight in which case it would be negative, in this problem, I don't min needs to be specified. So for example, finding the area under the curve from min (or 0) from 5.5 would give the probability of the boy gaining at least 5 pounds with continuity correction. –  Jeff Kwak Jan 16 '11 at 4:32
I just fixed some typos. Sorry about that. –  Jeff Kwak Jan 16 '11 at 4:36

You are correct that the problem needs Bayes' Theorem. It's hard to tell what mistake your teacher is making, but it looks like he is calculating the P-value: the probability that the boy's weight is that low, given the null hypothesis that he is taking his medication. This is a method for what is called "significance testing", but it certainly shouldn't be used in this problem, and definitely not in the way your professor has carried it out. In particular your teacher has made no use at all of the fact that 75% percent of patients do take their medication.

The method you've used to tackle the problem is entirely correct. Unfortunately you've made a small slip when you calculated P(B). Although you wrote at the start that:
P(B) = P(B|A)P(A) + P(B|not A)P(not A)
when you came to calculate it you used
P(B) = P(B|A) + P(B|not A)
so you got that P(B)=0.0151+0.0796=0.0947 and not P(B)=0.0151*0.75+0.0796*0.25=0.031225 If you correct this in your calculations your final answer goes up to 36.26% (even further away from your teacher's answer!)

As TonyK points out, the question tells us that the child claims to have taken the medicine. Really we should include this in our calculations as evidence (since the child is more likely to claim that he took the medicine if he did in fact take it). However it is hard to ascribe numerical probabilities to statements like "the boy is lying given that he took the medicine", so this would make the problem much harder. Of course, neither your teacher nor the original writer of the question expected you to take this piece of information into account.

-
I agree entirely with your last sentence! What this means is that the writer of the question was wrong to give us the information, because any additional information skews the probabilities. The patient should have been admitted to ER in a coma, so his veracity quotient wouldn't be a factor. It's almost as if the question write had never watched House MD. –  TonyK Jan 16 '11 at 15:50
Everybody lies. Sadly I doubt having him arrive in a coma would constitute less evidence about whether he'd been taking his meds or not. –  Oscar Cunningham Jan 16 '11 at 16:55
Damn, you're right. We need to have him arriving in ER because a crane fell on him. –  TonyK Jan 16 '11 at 20:33

This question is fatally flawed. To answer it correctly, you would need to know the probability that the boy would lie given that he skipped his shots. The question seems to assume that this probability is $1$.

-
And further that he would never lie by saying he had skipped his shots when in fact he had not. –  WilliamKF Sep 6 '12 at 16:42