Check your understanding with this (made up poblem): A portfolio manager claims his portfolio’s weekly return is 10% per week. You question that and take a sample of his returns for some past 30 weeks, and find (to your suprpise) that the sample’s mean is 10.05% with a standard deviation of 0.01%. Assume returns are normally distributed. Using a 95% confidence level, show H0 and Ha, and draw your own conclusion.

Joey, wait off on this one a little.

Actually, I just wanted to inquire about the copyright status of your problem. Can I use it or post it elsewhere on AF?

NO way, this is purely my own, and if you do you’re in deep violation of intellectual property laws, and most importantly in violation of AF laws.

“you’re in deep violation of intellectual property laws” the story of my life. Sigh.

I’d be interested in hearing about it in another thread entitled “Dear Abby”.

it’s early, but i’ll give it a wing. sample size 30, normally distributed, no population information but i’ll rely on the central limit theorem and choose the z test. Ho: mu<= 10% Ha: mu> 10% n=30 x-bar=10.05% s=0.01% mu=10% z(0.05) = (10.05%-10%)/(0.01%/sqrt(30)) = 27.39 > 1.645 therefor reject the null… PM is a stud again just winging it

What about using Student t, with two-tail test, and H0 = 10% and Ha different from 10%? The result is different to one decimal (27.38 > 2.045) and we still reject the H0. I’m under impression that we are testing to see if PM’s return is different from 10%, not necessarily greater than 10%.

I like the one tail test better. The t vs z thing is not important here because the samplle size is pretty large and the results are wildly signifcant. Usually though you would use a t test because you have a sample standard deviation (ie., the omniscient test maker has not told you the std. dev).

FisherSU, that’s correct. I think this is a good example with lessons to learn including: 1. He claims his return is 10% per week, so you set your H0 = 10%. 2. Why do you set H0=10% and not H0 <= 10% (or any other variety)? You are trying to disprove his claim. If you can show that his return is *not* equal 10%, +/- some standard deviation, then his claim is invalid. Your goal is to disprove his claim, no fancy footwork needed. 3. Not true that he means exactly 10%…there is variance in this whether stated or not. 4. If he doesn’t state his variance, then that’s why you use the sample’s variance as an approximation. 5. It’s a two tail test, why? The easiest way to remember this is to look at Ha: if it’s one directional then it is 1-tail. If it has 2 directions then it’s 2-tails. Ha != 0 has 2 directions. Ha > 9, has only 1 direction, etc. 6. Even though the sample’s mean is very close to his claim (10.05% vs 10%) we still rejected the claim, because that is still *very* far from 10%, as shown by t_calc. 7. You should try to see if his claim would hold if we have a sample mean= 10.01% and a std deviation of 0.01% (sounds close enough, doesn’t it?). It is still false, i.e., reject his claim. Homework problems: 1. What if he states that his return is 10% with variance of zero? How would that work? 2. What if he states his return as between 10-20%, how do you set H0 and Ha?