PARAMETRIC TESTS

Parametric test provide inferences for making statements about the means of parent populations.  A t test is often used for this purpose.  This test is based on the Student’s t statistic. 

 The t statistic assumes that the variable is normally distributed and the mean is known (or assume to be known) and the population variance is estimated from the sample.  An assumption about the normal distribution of the random variable is made with mean m and unknown population variance s2, which is estimated by the sample variance s2.

The t distribution is similar to the normal distribution in appearance.  However, both distributions are bell shaped and symmetric.  But, as compared to the normal distribution, the t distribution is more in the tails and less in the center.  This is because population variance s2 is unknown and is estimated by the sample variance s2. Given the uncertainty in the value of s2, the observed values of are more variable than those of z.  Thus, we must go a larger number of standard deviation from 0 to encompass a certain percentage of values from the t distribution than is the case with the normal distribution.  Yet, as the number of degrees of freedom increases, the t distribution reaches the normal.  In fact, for large samples of 120 or more, the t distribution and the normal distribution are virtually indistinguishable.  Table shows selected percentiles of the t distribution.  Although normally is assumed, the t test is quite robust to departures from normality.

The procedure for hypothesis testing for the special case when the t statistic is used is a follows.

1.      Formulate the null (H0) and the alternative (H1) hypotheses.

2.      Select the appropriate formula for the t statistic.

3.      Select a significance level,a for testing H0.Typically, the 0.05 levels are selected.

4.      Take one or two samples and compute the mean and standard deviation for each sample.

5.      Calculate the t statistic assuming H0 is true.

6.      Calculate the degrees of freedom and estimate the probability of getting a more extreme value of the statistic from table.  (Alternatively, calculate the critical value of the t statistics.)

7.      If the probability compute in step 6 is smaller than the significance level selected in step 3, reject H0. If the probability is larger, do not reject H0. (Alternately, if the value of he calculated statistic in step 5 is larger than the critical value determined in steep 6, reject H0. If the calculated value is smaller than the critical value, do not reject H0).  Failure to reject H0 does not necessarily imply that H0 is true. It only means that the true state is not significantly different than that assume by H0.

8.      Express the conclusion reached by the t test in terms of the marketing research problem.

One Sample

In marketing research, the researcher is often interested in making statements about a single variable against a known or given standard.  Examples of such statements include: the market share for a new product will exceed 15 percent, at least 65 percent of customers will like a new package design, and 80 percent of dealers will prefer the new pricing policy.  These statements can be translated to null hypotheses that can be tested using a one-sample test, such as the t test or the z test.  In the case of a t test for a single mean, the researcher is inclined to testing whether the population mean conforms to a given hypothesis (H0).

Two Independent Samples

Several hypotheses in marketing related to parameters from two different populations: for example, the users and nonusers of a brand differ in terms of their perceptions of the brand, the high-income consumers spend more on entertainment than low-income consumers, or the proportion of brand loyal users in segment I is more than the proportion in segments II.  Samples drawn randomly from different populations are termed independent samples.

Paired Samples

In many marketing research applications, the observations for the two groups are not selected from independent samples. Rather, the observations relate to paired samples in that the two sets of observations relate to the same respondents.  A sample of respondents may rate two competing brands, indicate the relative importance of two attributes of a product, or evaluate a brand at two different times. The difference in these cases is examined by a paired samples t test.  To compute t for paired samples, the paired difference variable, denoted by D, is formed and its mean and variance calculated.  Then the t statistic is computed.