Probability calculation

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 29 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 0.50.

 

What is the probability this could occur by chance?

Answer 1

It would appear that want to solve this using significance test for a proportion.

our z = (Ps - Po)/square root of (Po x(1-Po))/n Ps =29/34, observed proportion, Po = 0.5, proportion if theory is wrong. and n = 34, our sample size.

z = (29/34 - 17/34)/{square root of (0.5 x 0.5/34) = 12/34/square root of (0.25/34)

z = 0.353//square root of 0.007353)

z = 0.353/0.08585

z= 4.1

A z value this high basically puts the odds of January theory being wrong and the actual odds are 50-50 at near zero percent.