According to the \"January theory,\" if the stock market is up for the month of

ID: 3129438 • Letter: A

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 22 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?

Explanation / Answer

Note that P(at least x) = 1 - P(at most x - 1).

Using a cumulative binomial distribution table or technology, matching

n = number of trials =    34
p = the probability of a success =    0.5
x = our critical value of successes =    22

Then the cumulative probability of P(at most x - 1) from a table/technology is

P(at most   21   ) =    0.939275258

Thus, the probability of at least   22   successes is

P(at least   22   ) =    0.060724742 [ANSWER]

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Hi! I used binomial distirbution above. In case you use normal approximation instead, then:

We first get the z score for the critical value:

x = critical value =    21.5
u = mean = np =    17

s = standard deviation = sqrt(np(1-p)) =    2.915475947

Thus, the corresponding z score is

z = (x-u)/s =    1.543487266

Thus, the right tailed area is

P(z >   1.543487266   ) =    0.061356297    [ANSWER, using normal approximation]

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According to the \"January theory,\" if the stock market is up for the month of

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According to the \"January theory,\" if the stock market is up for the month of

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