Parametric tests provide inferences for making statements about the means of parent populations, At test is commonly used for this purpose. This test is based on the Student’s statistic, The t statistic assumes that the variable is normally distributed and the mean is known (or assumed to be known), and the population variance is estimated from the sample. Assume that the random variable X is normally distributed, with mean p. and unknown population variance (F2, which is estimated by the sample variance s2. Recall that the standard deviation of the sample mean, X, is estimated as sx = s/√n. Then t = (X – µ.) is t distributed with n – 1 degrees of freedom.
The t distribution is similar to the normal distribution in appearance, Both distributions are bell shaped and symmetric, However as compared to the normal distribution, the t distribution has more area in the tails and less in the center, This is because population variance (F2 is unknown and is estimated by the sample variance s2. Given the uncertainty in the value of s2, the observed values of t are more variable than those of z, Thus, we must go a larger number of standard deviations 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 approaches the normal distribution. In fact, for large samples of 120 or more, the distribution and the normal distribution are virtually indistinguishable. Table 4 in the Statistical’Appendix shows selected percentiles of the I distribution. Although normality is assumed, the 1 test is quite robust to departures from normality.
The procedure for hypothesis testing, for the special case when the statistic is used, is as follows.
1. Formulate the null (Ho and the alternative (H¹) hypotheses.
2. Select the appropriate formula for the t statistic.
3. Select a significance level, a, for testing Ho’ Typically, the 0.05 level is selected.’?
4. Take one or two samples and compute the mean and standard deviation for each sample.
5. Calculate the t statistic assuming Ho is true.
6. Calculate the degrees of freedom and estimate the probability of getting a more extreme value of the statistic from Table 4. (Alternatively, calculate the critical value of the I statistic.)
7. If the probability computed in step 6 is smaller than the significance level selected in step 3, reject Ho If the probability is larger, do not reject Ho’ (Alternatively, if the absolute value of the calculated statistic in step 5 is larger than the absolute critical value determined in step 6, reject Ho If the absolute calculated value is smaller than the absolute critical value, do not reject Ho) Failure to reject Ho does not necessarily imply that Ho is true. It only means that the true state is not significantly different from that assumed by Ho.14
8. Express the conclusion reached by the I test in terms of the marketing research problem.
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; 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 test or the z test. In the case of a test for a single mean, the researcher is interested in testing whether the population mean conforms to a given hypothesis (Ho)’ For the data in Table 15.1, suppose we wanted to test the hypothesis that the mean familiarity rating exceeds 4.0, the neutral value on a 7-point scale A significance level of a = 0.05 is selected. The hypotheses may be formulated as:
The degrees of freedom for the t statistics to test the hypothesis about one mean are n – 1, In this case, n – I = 29 – I or 28. From Table 4 in the Statistical Appendix, the probability of getting a more extreme value than 2.471 is less than 0.05. (Alternatively, the critical value for 28 degrees of freedom and a significance level of 0.05 is 1.7011, ‘which is less than the calculated value.) Hence, the null. hypothesis is rejected. The familiarity level does exceed 4.0. Note that if the population standard deviation was assumed to be known as 1.5, rather than estimated from the sample, a z test would be appropriate. In this case, the value of the z statistic would be:
Two Independent Samples
Several hypotheses in marketing relate 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 1 is more than the proportion in segment II. Samples drawn randomly from different populations are termed independent samples. As in the case for one sample. the hypotheses could relate to means or proportions.
MEANS In the case of means for two independent samples, the hypotheses take the following form.
The two populations are sampled and the means and variances computed based on samples of sizes n1 and n2. If both populations are found to have the same variance, a pooled variance estimate is computed from the two sample variances as follows:
The degrees of freedom in this case are (n1 + n2 – 2).
If the two populations have unequal variances. an exact t cannot be computed for the difference in sample means. Instead, an approximation to is computed. The number of degrees freedom in this case is usually not an integer, but a reasonably accurate probability can be obtained by rounding to the-nearest integer.
An F test of sample variance may be performed if it is not known whether the two populations have equal variance. In this case the hypotheses are:
The F statistic is computed from the sample variances as follows:
As can be seen, the critical value of the F distribution- depends upon two sets of degrees of freedom-those in the numerator and those in the denominator. The critical values of F for various degrees of freedom for the numerator and denominator are given in Table 5 of the Statistical Appendix. If the probability of F is greater than the significance level a, Ho is not rejected, and t based on the pooled variance estimate can be used. On the other hand, if the probability of F is less than or equal to a, Ho is rejected and t based on a separate variance estimate is used.
Using the data of Table 15.1, suppose we wanted to determine whether Internet usage was different for males as compared to females. A two-independent-samples t test was conducted. The results are presented in Table 15.14. Note that the F test of sample variances has a probability that is less than 0.05. Accordingly, Ho is rejected, and the t test based on “equal variances not assumed” should be used. The t value is -4.492 and, with 18.014 degrees of freedom, this gives a probability of 0.000, which is less than the significance level of 0.05. Therefore, the null hypothesis of equal means is rejected. Because the mean usage for males (sex = I) is 9333 and that for females (sex = 2) is 3.867, we conclude that males use the Internet to a significantly greater extent than females. We also show the t test assuming equal variances because most computer programs can automatically
conduct the test both ways. Instead of the small sample of 30, if this were a large and representative sample, there are profound implications for Internet service providers such as AOL, Earth Link, and the various telephone (e.g., Verizon) and cable (e.g., Comcast) companies. In order to target the heavy Internet users, these companies should focus on males. Thus, more advertising dollars should be spent on magazines that cater to male audiences than those that target females.
Stores Seek to Suit Elderly to a “t”
A study based on a national sample of 789 respondents who were age 65 or older attempted to determine the effect that lack of mobility has on patronage behavior. A major research question related to the differences in the physical requirements of dependent and self-reliant elderly persons. That is, did the two groups require different things to get to the store or after they arrived at the store? A more detailed analysis of the physical requirements conducted by two-independent-sample t tests (shown in the accompanying table) indicated that dependent elderly persons are more likely to look for stores that offer home delivery and phone orders, and stores to which they have accessible transportation. They are also more likely to look for a variety of stores located close together. Retailers, now more than ever, are realizing the sales potential in the elderly market. With the baby-boomer generation nearing retirement in 2010, stores such as WaLL-Mart, Cold water Creek, and Williams-Sunoma see “the icing on the cake.” The elderly shoppers are more likely to spend more money and become patrons of 3 store. However, to attract them, stores should offer home delivery and phone orders, and arrange accessible transportation. 16
Differences in Physical Requirements Between Dependent and Self-Reliant Elderly
PROPORTIONS The case involving proportions for two independent samples is also illustrated using the data of Table 15.1, which gives the number of males and females who use the Internet for shopping. Is the proportion of respondents using the Internet for shopping the same for males and females? The null and alternative hypotheses are:
A z test is used as in testing the proportion for one sample. However, in this case the test statistic is given by:
In the test statistic, the numerator is the difference between the proportions in the two samples, PI and P2 ‘The denominator is the standard error of the difference in the two proportions and is given by:
A significance level of a = 0.05 is selected. Given the data of Table 15,1. the test statistic can be calculated as:
Given a two-tail test. the area to the right of the critical value is a/2 or 0.025. Hence. the critical value of the test statistic is 1.96. Because the calculated value is less than the critical value. the null hypothesis cannot be rejected. Thus. the proportion of users (0.733) for males and (0.400) for females is not significantly different for the two samples, Note that although the difference is substantial. it is not statistically significant due to the small sample sizes (15 in each group).