Help me solve a probability question

[mhamini]

One medical company invented some pills that should accelerate the growth of hair. The company wanted to try the pills on people - and so 2 groups of volunteers (each had 100 persons) were chosen to participate in the test. Persons in 1st group didn't take the pills and the experts noticed that the average monthly increment of the length of their hair was 3,5 cm, person in the 2nd group took the pills and the experts found out that the average monthly icrement was 3,9 cm. It's known that the diffusibility (denoted D(X) for random variable X) of monthly increment of the length of hair is 0,5 cm^2. It was only a luck or the pills are really effective? What is the probability that by using the pills the hair will grow faster?

 

[steve10]

Re: Probability Q

Naturally, the monthly increments are taken as the random variables. Notice that in each group, we have 100 samples, we can use central limit theorem to conclude that the monthly increment random variable X₁ for group one follows the normal distribution N(3.5,((√(0.5))/(√(100)))), while the monthly increment random variable X₂ for group two follows N(3.9,((√(0.5))/(√(100)))). So, they have the same standard deviation but different means. Now, all you have to do is to calculate the probability

P(X₁-X₂<0)
= ∫∫_{x₁<x₂}f₁(x₁)f₂(x₂)dx₁dx₂.

Mathematica shows that the integral is equal to 0.999968.