Hypothesis Testing PowerPoint Presentation

Slide 1 -

HypothesisUnit IV

Slide 2 -

Hypothesis Hypothesis a proposition formulated for empirical testing; a tentative descriptive statement that describes the relationship between two or more variables. In classical tests of significance, two kinds of hypotheses are used. First null hypothesis (H0) A second, alternative hypothesis (HA)

Slide 3 -

Null hypothesis: A statement in which no difference or effect is expected. If the null hypothesis is not rejected, no changes will be made. Alternative hypothesis: A statement that some difference or effect is expected. Accepting the alternative hypothesis will lead to changes in opinions or actions.

Slide 4 -

Procedure for Hypothesis Testing The following steps are involved in hypothesis testing Formulate the null hypothesis H0 and the alternative hypothesis H1. Select an appropriate statistical technique and the corresponding test statistic. Choose the level of significance, . Determine the sample size and collect the data. Calculate the value of the test statistic. Determine the probability associated with the test statistic under the null hypothesis, using the sampling distribution of the test statistic. Alternatively, determine the critical values associated with the test statistic that divide the rejection and nonrejection regions. Compare the probability associated with the test statistic with the level of significance specified. Alternatively, determine whether the test statistic has fallen into the rejection or the nonrejection region. Make the statistical decision to reject or not reject the null hypothesis. Express the statistical decision in terms of the marketing research problem.

Slide 5 -

ppt slide no 5 content not found

Slide 6 -

Step 1: Formulate the Hypotheses The first step is to formulate the null and alternative hypotheses. A null hypothesis is a statement of the status quo, one of no difference or no effect. If the null hypothesis is not rejected, no changes will be made. An alternative hypothesis is one in which some difference or effect is expected. Accepting the alternative hypothesis will lead to changes in opinions or actions. Thus, the alternative hypothesis is the opposite of the null hypothesis. One is that the null hypothesis is rejected and the alternative hypothesis is accepted.

Slide 7 -

The other outcome is that the null hypothesis is not rejected based on the evidence. However, it would be incorrect to conclude that because the null hypothesis is not rejected, it can be accepted as valid. In classical hypothesis testing, there is no way to determine whether the null hypothesis is true. If the null hypothesis H0 is rejected, then the alternative hypothesis H1 will be accepted and the new Internet shopping service will be introduced. On the other hand, if H0 is not rejected, then the new service should not be introduced unless additional evidence is obtained.

Slide 8 -

Step 2: Select an Appropriate Test To test the null hypothesis, it is necessary to select an appropriate statistical technique. The researcher should take into consideration how the test statistic is computed and the sampling distribution that the sample statistic (e.g., the mean) follows. The test statistic measures how close the sample has come to the null hypothesis. The test statistic often follows a well-known distribution, such as the normal, t, or chi-square distribution. Guidelines for selecting an appropriate test or statistical technique are discussed later in this chapter. In our example, the z statistic, which follows the standard normal distribution, would be appropriate.

Slide 9 -

ppt slide no 9 content not found

Slide 10 -

Whenever we draw inferences about a population, there is a risk that an incorrect conclusion will be reached. Two types of errors can occur. TYPE I ERROR Type I error Also known as alpha error, it occurs when the sample results lead to the rejection of a null hypothesis that is in fact true. TYPE II ERROR Type II error Also known as beta error, it occurs when the sample results lead to the nonrejection of a null hypothesis that is in fact false.

Slide 11 -

Type II error occurs when, based on the sample results, the null hypothesis is not rejected when it is in fact false. In our example, the Type II error would occur if we concluded, based on sample data, that the proportion of customers preferring the new service plan was less than or equal to 0.40 when, in fact, it was greater than 0.40. The probability of Type II error is denoted by . Unlike , which is specified by the researcher, the magnitude of depends on the actual value of the population parameter (proportion). The probability of Type I error ( ) and the probability of Type II error ( ) are shown in Figure 15.4. The complement (1 - b) of the probability of a Type II error is called the power of a statistical test.

Slide 12 -

ppt slide no 12 content not found

Slide 13 -

POWER OF A TEST The power of a test is the probability (1 - β) of rejecting the null hypothesis when it is false and should be rejected. Although β is unknown, it is related to α. An extremely low value of (e.g., 0.001) will result in intolerably high β errors. So it is necessary to balance the two types of errors. As a compromise, α is often set at 0.05; sometimes it is 0.01; other values of are rare. The level of along with the sample size will determine the level of β for a particular research design. The risk of both α and β can be controlled by increasing the sample size. For a given level of α, increasing the sample size will decrease β, thereby increasing the power of the test.

Slide 14 -

Step 4: collect data and calculate test statistic Sample size is determined after taking into account the desired α and β errors and other qualitative considerations, such as budget constraints. Then the required data are collected and the value of the test statistic computed. In our example, 30 users were surveyed and 17 indicated that they used the Internet for shopping. Thus the value of the sample proportion is p = 17/30 = 0.567

Slide 15 -

ppt slide no 15 content not found

Slide 16 -

Step 5: determine the probability (or critical value) Using standard normal tables,, the probability of obtaining a z value of 1.88 can be calculated. The shaded area between - ∞ and 1.88 is 0.9699. Therefore, the area to the right of z 1.88 is 1.0000 0.9699 0.0301. This is also called the p value and is the probability of observing a value of the test statistic as extreme as, or more extreme than, the value actually observed, assuming that the null hypothesis is true. Alternatively, the critical value of z, which will give an area to the right side of the critical value of 0.05, is between 1.64 and 1.65 and equals 1.645. Note that in determining the critical value of the test statistic, the area in the tail beyond the critical value is either α or α/2. It is α for a one-tailed test and α/2for a two-tailed test.

Slide 17 -

Step 6 and 7compare the probability (or critical value) and make the decision The probability associated with the calculated or observed value of the test statistic is 0.0301. This is the probability of getting a p value of 0.567 when π = 0.40. This is less than the level of significance of 0.05. Hence, the null hypothesis is rejected. Alternatively, the calculated value of the test statistic z 1.88 lies in the rejection region, beyond the value of 1.645. Again, the same conclusion to reject the null hypothesis is reached. Note that the two ways of testing the null hypothesis are equivalent but mathematically opposite in the direction of comparison.

Slide 18 -

ppt slide no 18 content not found

Slide 19 -

Step 8: Marketing research conclusion The conclusion reached by hypothesis testing must be expressed in terms of the marketing research problem. In our example, we conclude that there is evidence that the proportion of Internet users who shop via the Internet is significantly greater than 0.40. Hence, the recommendation to the department store would be to introduce the new Internet shopping service. As can be seen from Figure 15.6, hypotheses testing can be related to either an examination of associations or an examination of differences. In tests of associations, the null hypothesis is that there is no association between the variables (H0: . . . is NOT related to . . . ). In tests of differences, the null hypothesis is that there is no difference (H0: . . . is NOT different from . . . ). Tests of differences could relate to distributions, means, proportions, medians, or rankings. First, we discuss hypotheses related to associations in the context of cross-tabulations.

Slide 20 -

Test of Hypotheses

Slide 21 -

Hypothesis tests related to association

Slide 22 -

Hypothesis-testing procedures can be broadly classified as parametric or nonparametric, based on the measurement scale of the variables involved. Parametric tests assume that the variables of interest are measured on at least an interval scale. Nonparametric tests assume that the variables are measured on a nominal or ordinal scale. These tests can be further classified based on whether one, two, or more samples are involved.

Slide 23 -

the number of samples is determined based on how the data are treated for the purpose of analysis, not based on how the data were collected. The samples are independent if they are drawn randomly from different populations. For the purpose of analysis, data pertaining to different groups of respondents, for example, males and females, are generally treated as independent samples. On the other hand, the samples are paired when the data for the two samples relate to the same group of respondents. The most popular parametric test is the t test, conducted for examining hypotheses about means. The t test could be conducted on the mean of one sample or two samples of observations. In the case of two samples, the samples could be independent or paired.

Slide 24 -

Nonparametric tests based on observations drawn from one sample include the Kolmogorov-Smirnov test, the chi-square test, the runs test, and the binomial test. In case of two independent samples, the Mann-Whitney U test, the median test, and the Kolmogorov-Smirnov two-sample test are used for examining hypotheses about location. These tests are nonparametric counterparts of the two-group t test. The chi-square test can also be conducted.

Slide 25 -

For paired samples, nonparametric tests include the Wilcoxon matched pairs signed-ranks test and the sign test. These tests are the counterparts of the paired t test. In addition, the McNemar and chi-square tests can also be used. Parametric as well as nonparametric tests are also available for evaluating hypotheses relating to more than two samples.

Slide 26 -

Hypothesis Testing Related to Differences

Slide 27 -

Parametric Tests Parametric tests provide inferences for making statements about the means of parent populations. A t test is commonly 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 assumed to be known), and the population variance is estimated from the sample. 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 is unknown and is estimated by the sample variance.

Slide 28 -

The procedure for hypothesis testing, for the special case when the t statistic is used, is as 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, , for testing H0. 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 H0 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 t statistic.)

Slide 29 -

7. If the probability computed in step 6 is smaller than the significance level selected in step 3, reject H0. If the probability is larger, do not reject H0. (Alternatively, if the absolute value of the calculated t statistic in step 5 is larger than the absolute critical value determined in step 6, reject H0. If the absolute calculated value is smaller than the absolute 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 from that assumed by H0.14 8. Express the conclusion reached by the t test in terms of the marketing research problem. 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 α = 0.05 is selected.

Slide 30 -

One sample 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 t test or the z test. In the case of a t test for a single mean, the researcher is interested in testing whether the population mean conforms to a given hypothesis (H0).

Slide 31 -

ppt slide no 31 content not found

Slide 32 -

The degrees of freedom for the t statistic to test the hypothesis about one mean are n 1. In this case, n 1 29 1 or 28. the probability of getting a more extreme value than 2.471 is less than 0.05. (Alternatively, the critical t 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:

Slide 33 -

ppt slide no 33 content not found

Slide 34 -

the probability of getting a more extreme value of z than 2.595 is less than 0.05. (Alternatively, the critical z value for a one-tailed test and a significance level of 0.05 is 1.645, which is less than the calculated value.) Therefore, the null hypothesis is rejected, reaching the same conclusion arrived at earlier by the t test. The procedure for testing a null hypothesis with respect to a proportion for a single sample was illustrated earlier in this chapter when we introduced hypothesis testing.

Slide 35 -

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 I 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.

Slide 36 -

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. The degrees of freedom are n 1, where n is the number of pairs. a paired t test could be used to determine if the respondents differed in their attitude toward the Internet and attitude toward technology.

Slide 37 -

Paired samples: In hypothesis testing, the observations are paired so that the two sets of observations relate to the same respondents. paired samples t test: A test for differences in the means of paired samples.

Slide 38 -

Nonparametric Tests Nonparametric tests are used when the independent variables are nonmetric. Like parametric tests, nonparametric tests are available for testing variables from one sample, two independent samples, or two related samples. One Sample Sometimes the researcher wants to test whether the observations for a particular variable could reasonably have come from a particular distribution, such as the normal, uniform, or Poisson distribution. Knowledge of the distribution is necessary for finding probabilities corresponding to known values of the variable or variable values corresponding to known probabilities.

Slide 39 -

Kolmogorov-Smirnov(K-S) one-sample test The Kolmogorov-Smirnov (K-S) one-sample test is one such goodness-of-fit test. The K-S compares the cumulative distribution function for a variable with a specified distribution. Ai denotes the cumulative relative frequency for each category of the theoretical (assumed) distribution, and Oi the comparable value of the sample frequency. The K-S test is based on the maximum value of the absolute difference between Ai and Oi. The test statistic is: K = Max | Ai - Oi |

Slide 40 -

The runs test is a test of randomness for the dichotomous variables. This test is conducted by determining whether the order or sequence in which observations are obtained is random. The binomial test is also a goodness-of-fit test for dichotomous variables. It tests the goodness of fit of the observed number of observations in each category to the number expected under a specified binomial distribution. For more information on these tests, refer to standard statistical literature.

Slide 41 -

Two Independent Samples When the difference in the location of two populations is to be compared based on observations from two independent samples, and the variable is measured on an ordinal scale, the Mann-Whitney U test can be used. This test corresponds to the two-independent-sample t test for interval scale variables, when the variances of the two populations are assumed equal. In the Mann-Whitney U test, the two samples are combined and the cases are ranked in order of increasing size. The test statistic, U, is computed as the number of times a score from sample 1 or group 1 precedes a score from group 2. If the samples are from the same population, the distribution of scores from the two groups in the rank list should be random.

Slide 42 -

An extreme value of U would indicate a nonrandom pattern, pointing to the inequality of the two groups. For samples of less than 30, the exact significance level for U is computed. For larger samples, U is transformed into a normally distributed z statistic. This z can be corrected for ties within ranks. We examine again the difference in the Internet usage of males and females. This time, though, the Mann-Whitney U test is used. A significant difference is found between the two groups, corroborating the results of the two-independent-samples t test reported earlier. Because the ranks are assigned from the smallest observation to the largest, the higher mean rank (20.93) of males indicates that they use the Internet to a greater extent than females (mean rank 10.07).

Slide 43 -

Two other independent-samples nonparametric tests are the median test and Kolmogorov- Smirnov test. The two-sample median test determines whether the two groups are drawn from populations with the same median. It is not as powerful as the Mann-Whitney U test because it merely uses the location of each observation relative to the median, and not the rank, of each observation. The Kolmogorov-Smirnov two-sample test examines whether the two distributions are the same. It takes into account any differences between the two distributions, including the median, dispersion, and skewness,

Slide 44 -

ppt slide no 44 content not found

Slide 45 -

Paired Sample An important nonparametric test for examining differences in the location of two populations based on paired observations is the Wilcoxon matched-pairs signed-ranks test. This test analyzes the differences between the paired observations, taking into account the magnitude of the differences. So it requires that the data are measured at an interval level of measurement. This test computes the differences between the pairs of variables and ranks the absolute differences. The next step is to sum the positive and negative ranks.

Slide 46 -

The example considered for the paired t test, whether the respondents differed in terms of attitude toward the Internet and attitude toward technology, is considered again. Suppose we assume that both these variables are measured on ordinal rather than interval scales. Accordingly, we use the Wilcoxon test.

Slide 47 -

ppt slide no 47 content not found

Slide 48 -

The results are shown in Table 15.18. Again, a significant difference is found in the variables, and the results are in accordance with the conclusion reached by the paired t test. There are 23 negative differences (attitude toward technology is less favorable than attitude toward Internet). The mean rank of these negative differences is 12.72. On the other hand, there is only one positive difference (attitude toward technology is more favorable than attitude toward Internet). The mean rank of this difference is 7.50. There are six ties, or observations with the same value for both variables. These numbers indicate that the attitude toward the Internet is more favorable than toward technology. Furthermore, the probability associated with the z statistic is less than 0.05, indicating that the difference is indeed significant.