TESTING DIRECTIONAL HYPOTHESES
A Quick Review of Hypotheses, Assumptions, etc.
This is an addendum to the coverage of hypothesis testing we have included in your textbook.
Note that we have said that marketing researchers often use marketing research to test
assumptions they have about the consequences of a proposed decision alternative. Business
decision makers often rely on assumptions they believe to be true when making a choice from
among various decision alternatives. For example, a restaurant owner assumes that she must
provide a certain amount of food on each entre’ for her customers to believe they have “value” in
eating at her restaurant. These assumptions, when we express them as statements, may be
thought of as hypotheses. We make decisions based upon our hypotheses about how the world
works until evidence tells us that our hypothesis may not be correct. When it is important that
our hypothesis is correct yet we are not confident in it, we may opt to collect some data to test
the hypothesis before we make an important decision. When this happens, we are doing
marketing research.
Non-Directional v Directional Hypotheses
In the book you learned how to test hypotheses of the variety: “X is equal to Y.” An example
would be “I believe consumers will prefer Auto Model A at an average of “5.0” on a preference
scale ranging from 1 to 7. We call these non-directional hypotheses because we are not
specifying a direction. However, in some cases, we like to make hypotheses of the variety: “X
is > Y” or “X is < Y.” For instance, the hypothesis could be that consumers will prefer Model A
with an average “greater than 5.0” on the 1-7 scale. When we do this, we are making a
directional hypothesis. We can test directional hypotheses but we must make certain we
consider some differences from non-directional hypothesis testing.
Directional Hypotheses About Means
Let us assume we believe an attitude among the public is important to us. Let’s just say that the
attitude we are interested in is the feeling toward “all electric cars.” (Obviously, if we are in the
automotive business this attitude would be important to us). Further, we have a hypothesis that
this attitude is “generally favorable.” But, we need to quantify what we mean by “generally
favorable.” If we think of this attitude measured on a 7-point scale ranging from “1” Very Much
Not In Favor” to “7” Very Much In Favor” we can operationalize our “generally favorable” to be
“Greater than 5.” Note we are not saying the population’s average attitude is 5 (a non-directional
hypothesis). Rather we are saying it is >5 (a directional hypothesis). Now, if we were testing
our directional hypothesis using the logic we learned for a non-directional hypothesis we would
want our z value to fall within ±1.96 in order to accept the hypothesis. (Recall that for a nondirectional hypothesis, if you get a z Value from your formula that falls within ±1.96 (for the
95% level of confidence) or within ±2.58 (for the 99% level of confidence), then we accept (or
fail to reject) our hypothesis). But, since we are testing a directional hypothesis our new critical
value for a z is >1.64. Why isn’t it the same as we learned for our non-directional hypothesis?
In a directional hypothesis we only consider one side of the normal distribution. For a >
hypothesis, we are only concerned with the right side and for a < hypothesis, we are only
concerned with the left side of the normal distribution. Also, when we are testing a directional
hypothesis, as in our example, the Null hypothesis is that “the average attitude score is 5.” If this
is true we should have 95% of the sample observations resulting in a z score that falls at 1.64 or
less. Why 1.64? Because 1.64 defines 95% of the area under the curve when we are only
dealing with one side of the normal distribution (greater than 1.64 defines 5%). For comparison,
the ±1.96 z scores define 2.5% on EACH end of the normal distribution for a total of 5% when
we are using the entire distribution to test a non-directional hypothesis.
Now, let’s assume we test our example directional hypothesis (the average attitude score is
greater than 5) and, after we gather our sample data, we find the sample mean to be 6.2 and the
calculated z value to be 2.8. Think of what this z value means. First, if we got a mean score on
our attitude BELOW 5 we could stop at that point. (Remember we are only using one side of the
normal distribution so if we have some value that falls on the “other” side, the data certainly do
not support our > hypothesis). But, what does it mean that we got a z value of 2.8? IF the Null
hypothesis is true and the real population attitude IS 5, 95% of the z values should fall at 1.64 or
less. Since our z value of 2.8 is greater than 1.64, there is only a 5% chance of supporting the
Null hypothesis and a 95% chance that our alternative hypothesis is correct. Therefore we would
SUPORT the directional hypothesis that the “attitude toward all-electric cars is greater than 5 on
a 1-7 scale.”
The Critical Values for z when Testing Directional Hypotheses
Our example above was for a “greater than” hypothesis and we assumed we wanted to test our
hypothesis at the 95% level of confidence. The critical values of z for both “greater than” and
“less than” hypotheses and for both the 95% and 99% levels of confidence are shown below:
Level of Confidence
95%
99%
Direction of Hypothesis
Greater than
Less than
Greater than
Less than
Critical z Value
+1.64
-1.64
+2.33
-2.33
How to Test Directional Hypotheses Using IBM® SPSS®
If you read the book about how to test non-directional hypotheses using IBM® SPSS®, you
already know how to test directional hypotheses. That’s right; the procedure is identical. But,
you need to know that when you access ANALYZE; COMPARE MEANS; ONE SAMPLE t
TEST, when you enter the TEST VALUE you DO NOT enter a < or > sign. SPSS is expecting
you to know what you are testing. So, following our example above, you would enter a “5” for
the TEST VALUE. (Do not enter a > sign; you understand when you enter the “5” that you are
really testing for the directional hypothesis “greater than 5.” Now, the only thing you have to
remember is the critical value of z (remember to read the t in the SPSS output). Assuming you
are using the 95% level of confidence, you would accept a > hypothesis IF the t value you get is
both a positive number and is GREATER than 1.64. So a +1.65 (or greater) would mean you
accept the hypothesis that the real attitude in the population is “greater than 5 on a 1-7 scale.”
If you have a < hypothesis, you would accept it if you got a negative t value that is LESS than a 1.64. So a -1.65 would mean you accept the hypothesis.
Directional Hypotheses About Percents
Thus far, we have been discussing testing directional hypotheses about means. Just as we
explained in the book for non-directional hypotheses, you can also test directional hypotheses
about percents but you cannot do this using IBM SPSS. You can, however, use the same
formula we showed you how to use in the book for non-directional hypotheses except you use
different critical values of z, as we have explained above, to determine whether or not you reject
or fail to reject your hypothesis. Just remember, that what we have said about testing hypotheses
about means also applies to hypothesis tests about percentages except you must calculate the
latter by hand instead of using IBM® SPSS®.