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.

What is the probability this could occur by chance? (You will probably
need a software package such as Excel or MINITAB.)

I don't have these.

I do not completely understand your question but the probability of
such independent events happening in either case is 50% without a bias
in any direction. Also, the hypothesis is based on sample of very
small size and hence is not even statistically significant.

I am not sure if this is of any use .... but just my views anyways ..

if there was no economic growth then its a simple combinatorial
calculation. ie whats the chance of getting 29 ups out of 34 trials. 
so there are 278,256 combinations (34!/29!5!) ways of getting 29 ups.
total space is 2^34 = 17179869184. divide the two and you get
1.61966E-05.

so its unlikely if there's no economic growth, so a january approach will work.

but lets assume economies grow 3 years out of 4.  we would then expect
34*(3/4) = 25 ups by random chance. so the only extra is 4 ups out of
the 9 remaining spaces.  how likely is this ?

we've already accounted for economic growth so its 9!/(4!5!) = 126 out
of a space of 2^9 = 512. 126/512 is about 25%. this is
indistinguishable from random chance. so its not going to work and
you'd be wasting your time with minitab.

a better use of your time is to do some basic finance theory.  there
is an overwhelming body of evidence for the semi strong efficient
market hypothesis (http://cbdd.wsu.edu/kewlcontent/cdoutput/TR505r/page4.htm).
this says prices impound all publicly available information.  one of
the results of this is that stock prices will look like brownian
motion (with drift).  so there's no trend, pattern or bias that you
can arbitrage out. the best strategy, confirmed by the university of
chicago, is buy and never sell.

can i have $7 please ?