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posited, and then a test is performed to see if the null hypothesis can be (1) rejected
in favor of the alternative or (2) not rejected.
In this particular case, our null hypothesis assumes that self-perceived risk pref-
erence is independent of a particular mutual fund selection. That suggests that an
investor’s self-description as an investor is not related to the mutual funds he pur-
chase, or more strongly stated, does not cause a purchase of a particular type of
mutual fund. If our test suggests otherwise, that is, the test leads us to reject the
null hypothesis , then we conclude that it is likely to be dependent (related).
This discussion may seem tedious, but if you do not have a firm understanding
of tests of hypothesis, then the remainder of the chapter will be very difficult, if
not impossible, to understand. Before we move on to the calculations necessary for
performing the test, the following summarizes the general procedure we have just
discussed:
1) an assumption ( null hypothesis) that the variables under consideration are
independent, or that they are not related, is made
2) an alternative assumption ( alternative hypothesis) relative to the null is made
that there is dependence between variables
3) the chi-square test is performed on the data contained in a contingency table to
test the null hypothesis
4) the results, a statistical calculation, will be used to attempt to reject the null
hypothesis
5) if the null is rejected, then this implies that the alternative is accepted; if the null
is not rejected, then the alternative hypothesis is rejected
The chi-square test is based on a null hypothesis that assumes independence of
relationships. If we believe the independence assumption, then the overall fraction
of investors in a perceived risk category and fund type should be indicative of the
entire investing population. Thus, an expected frequency of investors in each cell can
be calculated. We will have more to say about this later in the chapter. The expected
frequency, assuming independence, is compared to the actual (observed) and the
variation of expected to actual is tested by calculating a statistic, the
2 statistic (
χ
is the lower case Greek letter chi). The variation between what is actually observed
and what is expected is based on the formula that follows. Note that the calculation
squares the difference between the observed frequency and the expected frequency,
divides by the expected value, and then sums across the two dimensions of the i by
j contingency table:
χ
i j [(obs ij
2
expval ij ) 2
χ
=
/
expval ij ]
where:
obs ij =
frequency or count of observations in the ith row and jth column of the
contingency table
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