Probability Statistics

Question

According to the "January theory," if the stock market is up for the month of January, it will be up for the year. If it is down in January, it will be down for the year. According to an article in The Wall Street Journal, this theory held for 23 out of the last 34 years. Suppose there is no truth to this theory; that is, the probability it is either up or down is 0.5.



What is the probability this could occur by chance? (Round your answer to 6 decimal places.)

Answer 1

Hi Shawna,

This type of problem is usually solved with a binomial test.
The formula for this test is:

(X - p)/sqrt(p*q/n), where X is the proportion of times the theory worked, p is the proportion of times we expect it to work by chance, q = 1 - p, and n is the number of years that you tested the theory.

The first thing we have to do is compute proportion for the number of times it worked:
It worked 23 out of 34 times: 23/34 = .676471


Since we know the theory is false, we would it to work only 50% o the time or 0.5.
So the numerator of the formula becomes: .676471 - .5 = .176471

For the denominator, we know p. q is 1 - p = .5.
n is just the number of times you tested the theory, in your case, 34.

So the denominator is sqrt(.5*.5/34) = .085749

Now we find .176471/.085749 = 2.057983

 

This value is a z-score which we can use to find the probability. To do so, we need to use a normal distribution table (your teacher should have given you one or there should be one in your book). If we look up 2.057983 in the table, we find the value 0.019796. This number is the proportion of times you would expect the theory to work 23 or more times out of 34 tries (in other words, this should occur 1.9796% of the time).

 

Hope that helps.