© 2009 Pearson Education, Inc publishing as Prentice Hall 16-1 Chapter 16 Data Analysis: Frequency Distribution, Hypothesis Testing, and Cross-Tabulation

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-2 Figure 16.1 Relationship of Frequency Distribution, Hypothesis Testing, and Cross-Tabulation to the Previous Chapters and the Marketing Research Process Focus of This Chapter Relationship to Previous Chapters Relationship to Marketing Research Process Frequency General Procedure for Hypothesis Testing Cross Tabulation Research Questions and Hypothesis (Chapter 2) Data Analysis Strategy (Chapter 15) Problem Definition Approach to Problem Field Work Data Preparation and Analysis Report Preparation and Presentation Research Design

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-3 Technology Application to Contemporary Issues (Figs – 16.14) EthicsInternational Be a DM! Be an MR! Experiential Learning Opening Vignette What Would You Do? Fig Figure 16.2 Frequency Distribution, Hypothesis Testing, and Cross Tabulation: An Overview Frequency Distribution Statistics Associated With Frequency Distribution Introduction to Hypothesis Testing Cross Tabulation Statistics Associated With Cross Tabulation Cross Tabulation in Practice Tables Fig Tables Fig 16.5 Fig Fig Fig 16.11

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-4 Frequency Distribution In a frequency distribution, one variable is considered at a time. A frequency distribution for a variable produces a table of frequency counts, percentages, and cumulative percentages for all the values associated with that variable.

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-5 Calculate the Frequency for Each Value of the Variable Calculate the Percentage and Cumulative Percentage for Each Value, Adjusting for Any Missing Values Plot the Frequency Histogram Calculate the Descriptive Statistics, Measures of Location, and Variability Figure 16.3 Conducting Frequency Analysis

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-6

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-7 Table 16.2 Frequency Distribution of Attitude Toward Nike

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-8 Table 16.2 (Cont.) Frequency Distribution of Attitude Toward Nike Value LabelValueFrequencyPercentageValid Percentage Cumulative Percentage Very unfavorable Very favorable Missing Total

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-9 Figure 16.4 Frequency Histogram

© 2009 Pearson Education, Inc publishing as Prentice Hall Statistics Associated with Frequency Distribution Measures of Location The mean, or average value, is the most commonly used measure of central tendency. Where, X i = Observed values of the variable X n = Number of observations (sample size) The mode is the value that occurs most frequently. It represents the highest peak of the distribution.

© 2009 Pearson Education, Inc publishing as Prentice Hall The median of a sample is the middle value when the data are arranged in ascending or descending order. If the number of data points is even, the median is usually estimated as the midpoint between the two middle values – by adding the two middle values and dividing their sum by 2. The median is the 50th percentile. Statistics Associated with Frequency Distribution Measures of Location

© 2009 Pearson Education, Inc publishing as Prentice Hall The range measures the spread of the data. It is simply the difference between the largest and smallest values in the sample. Range = - Statistics Associated with Frequency Distribution Measures of Variability

© 2009 Pearson Education, Inc publishing as Prentice Hall The variance is the mean squared deviation from the mean. The variance can never be negative. The standard deviation is the square root of the variance. = = (-) 2 n-1  i 1 n Statistics Associated with Frequency Distribution Measures of Variability

© 2009 Pearson Education, Inc publishing as Prentice Hall Figure 16.6 A General Procedure for Hypothesis Testing Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Formulate H 0 and H 1 Select Appropriate Test Choose Level of Significance, α Collect Data and Calculate Test Statistic Determine Probability Associated with Test Statistic (TS CAL ) Determine Critical Value of Test Statistic TS CR Compare with Level of Significance, α Determine if TS CAL falls into (Non) Rejection Region Reject or Do Not Reject H 0 Draw Marketing Research Conclusion a) b) a)b)

© 2009 Pearson Education, Inc publishing as Prentice Hall A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis 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. The null hypothesis refers to a specified value of the population parameter (e.g., ), not a sample statistic (e.g., ).

© 2009 Pearson Education, Inc publishing as Prentice Hall A null hypothesis may be rejected, but it can never be accepted based on a single test. In classical hypothesis testing, there is no way to determine whether the null hypothesis is true. In marketing research, the null hypothesis is formulated in such a way that its rejection leads to the acceptance of the desired conclusion. The alternative hypothesis represents the conclusion for which evidence is sought. H 1 :  >0.40 A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis (Cont.)

© 2009 Pearson Education, Inc publishing as Prentice Hall The test of the null hypothesis is a one-tailed test, because the alternative hypothesis is expressed directionally. If that is not the case, then a two-tailed test would be required, and the hypotheses would be expressed as: H 0 :  =0.40 H 1 :  0.40 A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis (Cont.)

© 2009 Pearson Education, Inc publishing as Prentice Hall The test statistic measures how close the sample has come to the null hypothesis and follows a well-known distribution, such as the normal, t, or chi-square. In our example, the z statistic, which follows the standard normal distribution, would be appropriate. A General Procedure for Hypothesis Testing Step 2: Select an Appropriate Test where

© 2009 Pearson Education, Inc publishing as Prentice Hall Type I Error Type I error occurs when the sample results lead to the rejection of the null hypothesis when it is in fact true. The probability of type I error (  ) is also called the level of significance. Type II Error Type II error occurs when, based on the sample results, the null hypothesis is not rejected when it is in fact false. 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). A General Procedure for Hypothesis Testing Step 3: Choose a Level of Significance

© 2009 Pearson Education, Inc publishing as Prentice Hall Power of a Test The power of a test is the probability (1 -  ) of rejecting the null hypothesis when it is false and should be rejected. Although  is unknown, it is related to . An extremely low value of  (e.g., = 0.001) will result in intolerably high  errors. Therefore, it is necessary to balance the two types of errors. A General Procedure for Hypothesis Testing Step 3: Choose a Level of Significance (Cont.)

© 2009 Pearson Education, Inc publishing as Prentice Hall Figure 16.7 Type I Error (α) and Type II Error (β )

© 2009 Pearson Education, Inc publishing as Prentice Hall Figure 16.8 Probability of z With a One-Tailed Test Chosen Confidence Level = 95% Chosen Level of Significance, α=.05 z = 1.645

© 2009 Pearson Education, Inc publishing as Prentice Hall A General Procedure for Hypothesis Testing Step 4: Collect Data and Calculate Test Statistic In our example, the value of the sample proportion is p= 220/500 = The value of can be determined as follows: = =

© 2009 Pearson Education, Inc publishing as Prentice Hall The test statistic z can be calculated as follows: = = 1.83 A General Procedure for Hypothesis Testing Step 4: Collect Data and Calculate Test Statistic (Cont.)

© 2009 Pearson Education, Inc publishing as Prentice Hall Using standard normal tables (Table 2 of the Statistical Appendix), the probability of obtaining a z value of 1.83 can be calculated (see Figure 15.5). The shaded area between -  and 1.83 is Therefore, the area to the right of z = 1.83 is = 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 Note, in determining the critical value of the test statistic, the area to the right of the critical value is either  or  /2. It is  for a one-tail test and  /2 for a two-tail test. A General Procedure for Hypothesis Testing Step 5: Determine the Probability (Critical Value)

© 2009 Pearson Education, Inc publishing as Prentice Hall If the probability associated with the calculated or observed value of the test statistic (TS CAL ) is less than the level of significance (  ), the null hypothesis is rejected. The probability associated with the calculated or observed value of the test statistic is This is the probability of getting a p value of 0.44 when p = This is less than the level of significance of Hence, the null hypothesis is rejected. Alternatively, if the absolute calculated value of the test statistic |(TS CAL )| is greater than the absolute critical value of the test statistic |(TS CR )|, the null hypothesis is rejected. A General Procedure for Hypothesis Testing Steps 6 & 7: Compare the Probability (Critical Value) and Making the Decision

© 2009 Pearson Education, Inc publishing as Prentice Hall The calculated value of the test statistic z = 1.83 lies in the rejection region, beyond the value of 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. If the probability of TS CAL |TS CR | then reject H 0. A General Procedure for Hypothesis Testing Steps 6 & 7: Compare the Probability (Critical Value) and Making the Decision (Cont.)

© 2009 Pearson Education, Inc publishing as Prentice Hall 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 customers preferring the new plan is significantly greater than Hence, the recommendation would be to introduce the new service plan. A General Procedure for Hypothesis Testing Step 8: Marketing Research Conclusion

© 2009 Pearson Education, Inc publishing as Prentice Hall Figure 16.9 A Broad Classification of Hypothesis Testing Procedures Hypothesis Testing Test of AssociationTest of Difference MeansProportions

© 2009 Pearson Education, Inc publishing as Prentice Hall Cross-Tabulation While a frequency distribution describes one variable at a time, a cross-tabulation describes two or more variables simultaneously. Cross-tabulation results in tables that reflect the joint distribution of two or more variables with a limited number of categories or distinct values, e.g., Table 16.3.

© 2009 Pearson Education, Inc publishing as Prentice Hall Table 16.3 A Cross-Tabulation of Gender and Usage of Nike Shoes GENDER FemaleMaleRow Total Lights Users14519 Medium Users5510 Heavy Users51116 Column Total2421

© 2009 Pearson Education, Inc publishing as Prentice Hall Two Variables Cross-Tabulation Since two variables have been cross classified, percentages could be computed either columnwise, based on column totals (Table 16.4), or rowwise, based on row totals (Table 16.5). The general rule is to compute the percentages in the direction of the independent variable, across the dependent variable. The correct way of calculating percentages is as shown in Table 16.4.

© 2009 Pearson Education, Inc publishing as Prentice Hall Usage GENDER FemaleMale Light Users58.4%23.8% Medium Users20.8%23.8% Heavy Users20.8%52.4% Column Total100.0% Table 16.4 Usage of Nike Shoes by Gender

© 2009 Pearson Education, Inc publishing as Prentice Hall Usage GENDER FemaleMaleRaw Total Light Users73.7%26.3%100.0% Medium Users50.0% 100.0% Heavy Users31.2%68.8%100.0% Table 16.5 Gender by Usage of Nike Shoes

© 2009 Pearson Education, Inc publishing as Prentice Hall To determine whether a systematic association exists, the probability of obtaining a value of chi-square as large or larger than the one calculated from the cross-tabulation is estimated. An important characteristic of the chi-square statistic is the number of degrees of freedom (df) associated with it. That is, df = (r - 1) x (c -1). The null hypothesis (H 0 ) of no association between the two variables will be rejected only when the calculated value of the test statistic is greater than the critical value of the chi-square distribution with the appropriate degrees of freedom, as shown in Figure Statistics Associated with Cross-Tabulation Chi-Square

© 2009 Pearson Education, Inc publishing as Prentice Hall Figure Chi-Square Test of Association  2 Level of Significance, α Figure Chi- Square Test of Associ ation

© 2009 Pearson Education, Inc publishing as Prentice Hall Statistics Associated with Cross-Tabulation Chi-Square The chi-square statistic (  2 ) is used to test the statistical significance of the observed association in a cross-tabulation. The expected frequency for each cell can be calculated by using a simple formula: where= total number in the row = total number in the column = total sample size

© 2009 Pearson Education, Inc publishing as Prentice Hall Expected Frequency For the data in Table 16.3, for the six cells from left to right and top to bottom = (24 x 19)/45 = 10.1 = (21 x 19)/45 =8.9 = (24 x 10)/45 = 5.3 = (21 x 10)/45 =4.7 = (24 x 16)/45 = 8.5 = (21 x 16)/45 =7.5

© 2009 Pearson Education, Inc publishing as Prentice Hall = ( ) 2 + (5 – 8.9) (5 – 5.3) 2 + (5 – 4.7) (5 – 8.5) 2 + (11 – 7.5) = = 6.33

© 2009 Pearson Education, Inc publishing as Prentice Hall The chi-square distribution is a skewed distribution whose shape depends solely on the number of degrees of freedom. As the number of degrees of freedom increases, the chi- square distribution becomes more symmetrical. Table 3 in the Statistical Appendix contains upper-tail areas of the chi-square distribution for different degrees of freedom. For 2 degrees of freedom, the probability of exceeding a chi-square value of is For the cross-tabulation given in Table 16.3, there are (3-1) x (2-1) = 2 degrees of freedom. The calculated chi-square statistic had a value of Since this is greater than the critical value of 5.991, the null hypothesis of no association is rejected indicating that the association is statistically significant at the 0.05 level. Statistics Associated with Cross-Tabulation Chi-Square

© 2009 Pearson Education, Inc publishing as Prentice Hall The phi coefficient (  ) is used as a measure of the strength of association in the special case of a table with two rows and two columns (a 2 x 2 table). The phi coefficient is proportional to the square root of the chi-square statistic: It takes the value of 0 when there is no association, which would be indicated by a chi-square value of 0 as well. When the variables are perfectly associated, phi assumes the value of 1 and all the observations fall just on the main or minor diagonal. Statistics Associated with Cross-Tabulation Phi Coefficient

© 2009 Pearson Education, Inc publishing as Prentice Hall While the phi coefficient is specific to a 2 x 2 table, the contingency coefficient (C) can be used to assess the strength of association in a table of any size. The contingency coefficient varies between 0 and 1. The maximum value of the contingency coefficient depends on the size of the table (number of rows and number of columns). For this reason, it should be used only to compare tables of the same size. Statistics Associated with Cross-Tabulation Contingency Coefficient

© 2009 Pearson Education, Inc publishing as Prentice Hall Cramer's V is a modified version of the phi correlation coefficient, , and is used in tables larger than 2 x 2. or Statistics Associated with Cross-Tabulation Cramer’s V

© 2009 Pearson Education, Inc publishing as Prentice Hall Cross-Tabulation in Practice While conducting cross-tabulation analysis in practice, it is useful to proceed along the following steps: *Test the null hypothesis that there is no association between the variables using the chi-square statistic. If you fail to reject the null hypothesis, then there is no relationship. *If H 0 is rejected, then determine the strength of the association using an appropriate statistic (phi-coefficient, contingency coefficient, or Cramer's V), as discussed earlier. *If H 0 is rejected, interpret the pattern of the relationship by computing the percentages in the direction of the independent variable, across the dependent variable. Draw marketing conclusions.

© 2009 Pearson Education, Inc publishing as Prentice Hall Construct the Cross-Tabulation Data Calculate the Chi-Square Statistic, Test the Null Hypothesis of No Association Reject H 0 ? Interpret the Pattern of Relationship by Calculating Percentages in the Direction of the Independent Variable No Association Determine the Strength of Association Using an Appropriate Statistic Figure Conducting Cross-Tabulation Analysis NO YES

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-46

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-47

© 2009 Pearson Education, Inc publishing as Prentice Hall 16-48

© 2009 Pearson Education, Inc publishing as Prentice Hall SPSS Windows: Frequencies The main program in SPSS is FREQUENCIES. It produces a table of frequency counts, percentages, and cumulative percentages for the values of each variable. It gives all of the associated statistics. If the data are interval scaled and only the summary statistics are desired, the DESCRIPTIVES procedure can be used. The EXPLORE procedure produces summary statistics and graphical displays, either for all of the cases or separately for groups of cases. Mean, median, variance, standard deviation, minimum, maximum, and range are some of the statistics that can be calculated.

© 2009 Pearson Education, Inc publishing as Prentice Hall To select these frequencies procedures, click the following: Analyze > Descriptive Statistics > Frequencies... Or Analyze > Descriptive Statistics > Descriptives... orAnalyze > Descriptive Statistics > Explore... We illustrate the detailed steps using the data of Table SPSS Windows: Frequencies

© 2009 Pearson Education, Inc publishing as Prentice Hall SPSS Detailed Steps: Frequencies 1. Select ANALYZE on the SPSS menu bar. 2.Click DESCRIPTIVE STATISTICS, and select FREQUENCIES. 3.Move the variable "Attitude toward Nike " to the VARIABLE(s) box. 4.Click STATISTICS. 5.Select MEAN, MEDIAN, MODE, STD. DEVIATION, VARIANCE, and RANGE. 6.Click CONTINUE. 7.Click CHARTS. 8.Click HISTOGRAMS, then click CONTINUE. 9.Click OK.

© 2009 Pearson Education, Inc publishing as Prentice Hall SPSS Windows: Cross-tabulations The major cross-tabulation program is CROSSTABS. This program will display the cross-classification tables and provide cell counts, row and column percentages, the chi-square test for significance, and all the measures of the strength of the association that have been discussed. To select these procedures, click the following: Analyze > Descriptive Statistics > Crosstabs We illustrate the detailed steps using the data of Table 16.3.

© 2009 Pearson Education, Inc publishing as Prentice Hall SPSS Detailed Steps: Cross-Tabulations 1. Select ANALYZE on the SPSS menu bar. 2.Click DESCRIPTIVE STATISTICS, and select CROSSTABS. 3.Move the variable "User Group " to the ROW(S) box. 4.Move the variable "Sex " to the COLUMN(S) box. 5.Click CELLS. 6.Select OBSERVED under COUNTS, and select COLUMN under PERCENTAGES. 7.Click CONTINUE. 8.Click STATISTICS. 9.Click CHI-SQUARE, PHI, AND CRAMER'S V. 10.Click CONTINUE. 11.Click OK.

© 2009 Pearson Education, Inc publishing as Prentice Hall Excel: Frequencies The Tools > Data Analysis function computes the descriptive statistics. The output produces the mean, standard error, median, mode, standard deviation, variance, range, minimum, maximum, sum, count, and confidence level. Frequencies can be selected under the histogram function. A histogram can be produced in bar format. We illustrate the detailed steps using the data of Table 16.1.

© 2009 Pearson Education, Inc publishing as Prentice Hall Excel Detailed Steps: Frequencies 1.Select Tools (Alt + T). 2.Select Data Analysis under Tools. 3.The Data Analysis Window pops up. 4.Select Histogram from the Data Analysis Window. 5.Click OK. 6.The Histogram pop-up window appears on screen. 7.The Histogram window has two portions: a.Input b.Output Options:

© 2009 Pearson Education, Inc publishing as Prentice Hall Excel Detailed Steps: Frequencies (Cont.) 8.Input portion asks for two inputs. a.Click in the input range box and select (highlight) all 45 rows under ATTITUDE. $D$2: $D$46 should appear in the input range box. b.Do not enter anything into the Bin Range box. (Note that this input is optional; if you do not provide a reference, the system will take values based on range of the data set). 9.In the Output portion of pop-up window, select the following options: a.New Workbookb.Cumulative Percentage c.Chart Output 10.Click OK.

© 2009 Pearson Education, Inc publishing as Prentice Hall Excel: Cross-Tabulations The Data > Pivot Table function performs cross- tabulations in Excel. To do additional analysis or customize data, select a different summary function, such as maximum, minimum, average, or standard deviation. In addition, a custom calculation can be selected to analyze values based on other cells in the data plane. ChiTest can be assessed under the Insert > Function > Statistical > ChiTest function. We illustrate the detailed steps using the data of Table 16.3

© 2009 Pearson Education, Inc publishing as Prentice Hall Excel Detailed Steps: Cross-Tabulations 1.Select Data (Alt + D). 2.Under Data, select Pivot Table and Pivot Chart Wizard. 3.The Pivot Table and Pivot Chart Report window pops up. 4.Step 1 of 3: Pivot Table and Pivot Chart Wizard, leave as it is. 5.Click Next. 6.Step 2 of 3: Pivot Table and Pivot Chart Wizard. In the range box, select data under the columns CASENO, USERGR, SEX, ATTITUDE. $A$1:$D$46 should appear in the range box. Click Next.

© 2009 Pearson Education, Inc publishing as Prentice Hall Excel Detailed Steps: Cross-Tabulations (Cont.) 7.Step 3 of 3: Pivot Table and Pivot Chart Wizard, select New Worksheet in this window. Click Layout and drag the variables in this format. SEX | USERGR|CASENO (Double-click CASENO and select Count) | 8.Click OK and then Finish.

© 2009 Pearson Education, Inc publishing as Prentice Hall Exhibit 16.1 Other Computer Programs: Frequencies SAS The main program in SAS is UNIVARIATE. In addition to providing a frequency table, this program provides all of the associated statistics. Another procedure available is FREQ. For one-way frequency distribution, FREQ does not provide any associated statistics. If only summary statistics are desired, procedures such as MEANS, SUMMARY, and TABULATE can be used. It should be noted that FREQ is not available as an independent program in the microcomputer version. MINITAB The main function is Stats > Descriptive Statistics. The output values include the mean, median, mode, standard deviation, minimum, maximum, and quartiles. A histogram in a bar chart or graph can be produced from the Graph > Histogram selection.

© 2009 Pearson Education, Inc publishing as Prentice Hall Exhibit 16.2 Other Computer Programs: Cross-Tabulations SAS Cross-tabulation can be done by using FREQ. This program will display the cross-classification tables and provide cell counts as well as row and column percentages. In addition, the TABULATE program can be used for obtaining cell counts and row and column percentages, although it does not provide any of the associated statistics. MINITAB In Minitab, cross-tabulations and chi-square are under the Stats > Tables function. Each of these features must be selected separately under the Tables function.