Introduction to Marketing Research CHAPTERS 17 DATA ANALYSIS: Testing for Mean Differences Idil Yaveroglu Lecture Notes

Test the observed value of the mean for a single variable against a standard One-sample T-test

One-sample t-test The distribution of a t statistic is similar to the distribution of a z statistic in the assumption of normality; both are bell-shaped and symmetric. A univariate statistical technique used to test the observed mean for one variable against a known or given standard Because this test is focused on testing mean values, it can only be used with interval or ratio scaled data.

One-sample T-test Examples include: Does the average monthly household expenditure on groceries exceed $500? Monthly expenditures on grocery measured using a ratio scale Across the entire sample of respondents, is the average brand attitude toward Nike greater than 3.5? Nike brand attitude measured using an interval scale from 1 = very negative to 7 = very positive Has the mean preference level for visiting Disney parks (before entering the park) significantly increased or decreased since the last time the survey was conducted, where we found that the mean preference level was equal to 5.25? Preference level for visiting Disney measured using an interval scale from 1 = not at all preferred to 10 = very much preferred

One-sample T-test In a one-sample t-test, we test whether the mean value of a particular variable is different, or greater than or less than a given standard. For example, we can test whether the mean preference level for visiting Disney parks is significantly different from 5.25 Null hypothesis is the hypothesis of no difference No, there is no significant difference between the mean preference level in this sample and 5.25 Alternative hypothesis is the statement of a difference (greater than or less than) Yes, there is a significant difference in mean preference level in this sample from 5.25 A t-statistic (t) is used to test the significance of the difference.

Steps for Conducting a One-sample t-test Write out the null and alternative hypotheses Calculate the t-statistic Use the significance level corresponding to the t statistic to determine whether or not there is support for a significant difference between the observed mean and the given standard. If the significance level is <.05 then you can reject the null hypothesis and conclude that there is a significant difference If the significance level is >.05 then you are unable to reject the null hypothesis and conclude that there is no significant difference between the observed mean and the given standard Interpret the difference (or no difference) by discussing the observed mean reported in the one-sample statistic table in comparison to the test value

SPSS EXAMPLE: One-sample t-test RQ: Is average preference before entering the park significantly different from 5.25? Ho: Average preference is not significantly different from 5.25 H1: Average preference is significantly different (greater or less) than 5.25 Case number (Respondent ID) Sample 1 = teenagers 2 = adults Pref1 (Preference before visiting the park) Measured using a 10-point semantic differential scale where 1 = not at all preferred and 10 = very much preferred Pref2 (Preference after visiting the park)

SPSS EXAMPLE: One-sample t-test Select ANALYZE Click on COMPARE MEANS Select ONE-SAMPLE T-TEST Choose the “test variable” Select “Preference before visiting” Enter the “test value” Enter 5.25 as the test value Click on OK

SPSS EXAMPLE: One-sample t-test 1.73 < 2.093 or .1 > .025 Is there a significant difference in the average preference level for this sample and 5.25??? Although mean preference level has decreased since the last survey from 5.25 to 4.75 measured on a 10-point scale, this difference is not significant since the t-statistic (t=-1.730) does not exceed the critical value (critical value = 2.093 for a two-tailed test). Likewise the significance level (.10) is > .025. Thus, we cannot reject the null hypothesis and conclude that the average preference level is not significantly different (greater than or less than) 5.25.

Math: One-sample t-test Step 1: State the null and alternative hypotheses Step 2: Assume the null hypothesis is true Step 3: Compute a relevant test statistic Step 4: Degrees of freedom df= n-1 Step 5: Compare observed t to a t-table with df and appropriate significance level

Test the observed value of a proportion for a single variable against a standard One-sample z-test aka “Binomial test”

One-sample z-test A univariate statistical technique used to test the observed value of a proportion against a known or given standard Because this test is focused on percentages, it can be performed with data measured using any scale of measurement (nominal, ordinal, interval or ratio).

One-sample z-test Examples include: A sample of 100 people are asked whether they think an airline should paint its new planes red or blue. 43% said blue. The management wants to know if this is significantly different from 50%, i.e., different from chance? Did the percentage of respondents that ranked Brand X as #1 compared to the other brands in the list exceed 30%? Did the percentage of respondents that described themselves as “light users” exceed 20%?

One-sample z-test In a one-sample z-test, we test whether an observed proportion corresponding to responses to a particular variable is different from, or greater than or less than, a given standard For example, we can test whether the observed percentage of light users is greater than 20%. Null hypothesis is the hypothesis of not being significantly greater No, the proportion of light users in this sample is not greater than 20%. Alternative hypothesis is the statement of being significantly greater Yes, the proportion of light users in this sample is greater than 20%. A z-statistic (z) is used to test the significance of the difference.

Steps for Conducting a One-sample z-test Write out the null and alternative hypotheses Calculate the z-statistic Use the significance level corresponding to the z statistic to determine whether or not there is support for a significant difference between the observed and given proportion If the significance level is <.05 then you can reject the null hypothesis and conclude that there is a significant difference If the significance level is >.05 then you are unable to reject the null hypothesis and conclude that there is no significant difference between the observed proportion and the given standard Interpret the difference (or no difference) by discussing the observed proportion in comparison to the given proportion

SPSS EXAMPLE: One-sample z-test RQ: Is the percentage of light users greater than 20%? Ho: Percentage is not greater than 20% H1: Percentage is greater than 20% Case number = Respondent ID User group Coded as 1 for light users, 2 for medium users, 3 for heavy users Sex Coded as 1 for females, 2 for males Overall attitude towards Nike shoes Measured using a 7-point semantic differential scale coded as “punch the number circled” where 1 = very unfavorable and 7 = very favorable

SPSS EXAMPLE: One-sample z-test To conduct this type of test using SPSS, we can only examine variables that are binomial, i.e., have only two possible responses. In this case we must recode the variable “user group” so that 1 represents light users (our group of interest) and 2 represents both medium & heavy users (our contrast group) Then we can test the proportion of 1 against 2 (light users vs. everyone else) to determine if this proportion is greater than 15% Do this using the TRANSFORM → RECODE INTO DIFFERENT VARIABLES feature in SPSS

SPSS EXAMPLE: One-sample z-test Select ANALYZE Select NONPARAMETRIC TESTS Select LEGACY DIALOGUES Click on BINOMIAL Choose the “test variable” (this is the variable we just recoded) Select “r_user group” Enter the “test proportion” Type in (.20) Enter the “cut point” value to split the two groups 1.5 Click on OK

SPSS EXAMPLE: One-sample z-test .001<.05 Is the proportion of light users in the sample greater than 20%??? The observed proportion of light users in the sample is equal to 40%. Since the significance level (.001) is < .05 we can conclude that this is significantly greater than the test proportion which we set as 20. Thus, we can reject the null hypothesis and conclude that the observed proportion of light users (40%) is significantly greater than 20%.

Math: One-sample z-test Step 1: State the null and alternative hypotheses Step 2: Assume the null hypothesis is true Step 3: Compute a relevant test statistic Step 5: Compare observed z to critical z score with appropriate significance level

What Hypothesis Tests Will We Learn? Test associations between two discrete variables Crosstabs and Chi-Squared test we will do this today Test observed values of means or proportions of a single variable against a standard One sample t-test and One sample z-test we will do this today Test observed values of means across two variables Paired samples t-test Test differences across two or more groups of respondents Independent samples t-test and ANOVA Test associations between two scaled variables Correlation Test the significance of regression coefficients that correspond to predictors in a regression model Regression

Compare the mean across two different variables Paired samples T-test

Paired samples T-test A multivariate statistical technique used to test the difference in the mean value across two different variables Because this test is focused on testing mean values, it can only be used with interval or ratio scaled data. Because the test compares means across two different variables, these variables should be measured in the same way (i.e., using the same set of possible response options)

Paired samples T-test Examples include: Did the respondents agree more that the cashier was courteous than professional? Opinions for both variables measured using an interval scale from 1 = strongly disagree to 5 = strongly agree Do shoppers consider brand name to be more important than price when purchasing fashion clothing? Importance of both attributes measured using an interval scale from 1 = not at all important to 7 = very important Do households report spending more money on groceries compared to eating out per month? Money spent measured as a $ amount (ratio scale) Is the mean preference level for visiting Disney after going into the park different from the mean preference level before going into the park Preference level for visiting Disney both before and after entering the park measured using an interval scale from 1 = not at all preferred to 10 = very much preferred

Paired samples T-test In a paired samples t-test, we test whether the observed mean values for two variables are different, or that one mean value is greater than or less than another mean value For example, we can test whether the mean preference level after visiting the park is significantly different from the mean preference level before visiting the park Null hypothesis is the hypothesis of no difference No, there is no significant difference between the two mean preference levels Alternative hypothesis is the statement of a difference Yes, there is a significant difference between the two mean preference levels A t-statistic (t) is used to test the significance of the difference.

Steps for Conducting a Paired samples t-test Write out the null and alternative hypotheses Calculate the t-statistic Use the significance level corresponding to the t statistic to determine whether or not there is support for a significant difference between the two observed means If the significance level is <.05 then you can reject the null hypothesis and conclude that there is a significant difference If the significance level is >.05 then you are unable to reject the null hypothesis and conclude that there is no significant difference between the two observed means Interpret the difference (or no difference) by discussing the two observed means for the two variables

SPSS EXAMPLE: Paired samples t-test RQ: Is the mean preference level for Disney parks before entering the park different from the mean preference level upon leaving the park? Ho: Average preference level after visiting the park is not significantly different from the average preference level before visiting the park H1: Average preference level after visiting the park is significantly different from the average preference level before visiting the park Case number (Respondent ID) Sample 1 = teenagers 2 = adults Pref1 (Preference before visiting the park) Measured using a 10-point semantic differential scale where 1 = not at all preferred and 10 = very much preferred Pref2 (Preference after visiting the park)

SPSS EXAMPLE: Paired samples t-test Select ANALYZE Click on COMPARE MEANS Select PAIRED SAMPLES T-TEST Choose the two “paired variables” (NOTE: you can run this test for multiple pairs of variables simultaneously) Select “Preference before visiting” as variable 1 Select “Preference after visiting” as variable 2 Click on OK

SPSS EXAMPLE: Paired samples t-test Is there a significant difference in the average preference level before & after going to the park??? The t-statistic is equal to -14.873 with a corresponding significance level of .000. Since 14.873 exceeds the critical value of 2.093 and the significance level is <.025, we can reject the null hypothesis that there is no significant difference between the two mean values and conclude that the mean preference levels before and after visiting the Disney park are significantly different. Specifically, the mean level of preference before visiting in this sample is equal to 4.75 (measured on a 10-point scale) whereas the mean level of preference after visiting is equal to 7.70 (also measured on a 10-point scale). Thus, preference increased after visiting the park.

Math: Paired samples t-test Step 1: State the null and alternative hypotheses Step 2: Assume the null hypothesis is true Step 3: Compute a relevant test statistic Step 4: Degrees of freedom df = n-1 Step 5: Compare observed t to a t-table with df and appropriate significance level

What Hypothesis Tests Will We Learn? Test associations between two discrete variables Crosstabs and Chi-Squared test Test observed values of means or proportions of a single variable against a standard One sample t-test and One sample z-test Test observed values of means across two variables Paired samples t-test Test differences across two or more groups of respondents Independent samples t-test and ANOVA we will do this today Test associations between two scaled variables Correlation we will do this in the last class Test the significance of regression coefficients that correspond to predictors in a regression model Regression we will do this in the last class

Compare the mean across two different groups of respondents Independent samples T-test

Independent samples T-test A multivariate statistical technique used to test the difference in the mean value across two different groups of respondents Because this test is focused on testing mean values, it can only be used with interval or ratio scaled data. The groups that are being compared should be distinguished by a nominal or ordinal variable The variable that is being compared (in terms of its mean value) should be the same across both groups

Independent samples T-test Examples include: Did the airline travelers from Detroit (n=20) have a different mean satisfaction level with their flight compared to the airline travelers from Miami (n=20)? Satisfaction assessed using an interval scale from 1 = highly dissatisfied to 5 = highly satisfied Do the males and female users of a brand differ in terms of their attitude toward the brand? Brand attitude measured using an interval scale from 1= negative to 7 = positive Do high-income consumers spend more money on entertainment per month compared to low-income consumers? Money spent measured as a $ amount (ratio scale) Before entering a Disney park, do teenagers and adults differ in terms of their preference for visiting the Disney park? Preference level for visiting Disney both before entering the park measured using an interval scale from 1 = not at all preferred to 10 = very much preferred

Independent samples T-test In an independent samples t-test, we test whether the observed mean values for the same variable are different between two groups For example, we can test whether the mean preference level before visiting the park is significantly different for teenagers vs. adults Null hypothesis is the hypothesis of no difference No, there is no significant difference in the mean preference levels for teenagers vs. adults Alternative hypothesis is the statement of a difference Yes, there is a significant difference between the mean preference levels for teenagers vs. adults A t-statistic (t) is used to test the significance of the difference.

Steps for conducting an independent samples t-test Write out the null and alternative hypotheses Calculate the t-statistic Use the significance level corresponding to the t statistic to determine whether or not there is support for a significant difference between the two groups If the significance level is <.05 then you can reject the null hypothesis and conclude that there is a significant difference If the significance level is >.05 then you are unable to reject the null hypothesis and conclude that there is no significant difference between the two groups in terms of the observed means Interpret the difference (or no difference) by discussing the two observed means for the two groups

SPSS EXAMPLE: Independent samples t-test RQ: Does the mean preference level for Disney parks before entering the park differ for teenagers vs. adults? Ho: Average preference level before visiting the park is not significantly different among teenagers vs. adults H1: Average preference level before visiting the park is significantly different among teenagers vs. adults Case number (Respondent ID) Sample 1 = teenagers 2 = adults Pref1 (Preference before visiting the park) Measured using a 10-point semantic differential scale where 1 = not at all preferred and 10 = very much preferred Pref2 (Preference after visiting the park)

SPSS EXAMPLE: Independent samples t-test Select ANALYZE Click on COMPARE MEANS Select INDEPENDENT SAMPLES T-TEST Choose the test variable “Preference before visiting” Define the grouping variable by entering the codes that distinguish the two groups of respondents Enter 1 for Group 1 (teenagers) Enter 2 for Group 2 (adults) Click on CONTINUE Click on OK

SPSS EXAMPLE: Independent samples t-test Note that although the test for equality of variances is important, we will not discuss this in this class. Assume equal variances. Is there a significant difference in the average preference level before going into the park for teenagers vs. adults??? The t-statistic is equal to 3.143 with a corresponding significance level of .006. Since 3.143 exceeds the critical value of 2.101 and the significance level is <.025, we can reject the null hypothesis that there is no significant difference between the two mean values across the groups and conclude that the mean preference levels are significantly different for teenagers and adults. Specifically, we can see that teenagers reported a higher preference level before entering the park compared to adults. The mean preference level among teenagers is equal to 5.5 (measured on a 10-point scale) whereas the mean preference level among adults is equal to 4.0 (measured on a 10-point scale).

Math: Independent samples t-test Step 1: State the null and alternative hypotheses Step 2: Assume the null hypothesis is true Step 3: Compute a relevant test statistic Step 4: Degrees of freedom df = n1+n2-2 Step 5: Compare observed t to a t-table with df and appropriate significance level

Compare the mean across three or more groups of respondents One-way Analysis of Variance aka “ANOVA”

One-way Analysis of Variance (ANOVA) A multivariate statistical technique used to test the difference in the mean value across three or more groups of respondents Because this test is focused on testing mean values, it can only be used with interval or ratio scaled data. The groups that are being compared should be distinguished by a nominal or ordinal variable The variable that is being compared (in terms of its mean value) should be the same across all groups

One-way ANOVA Examples include: Three groups of passengers from Chicago, Miami, and Detroit all rated satisfaction scores on a 5 point scale. Do the mean satisfaction scores differ depending on what city the data was collected? Satisfaction assessed using an interval scale from 1 = highly dissatisfied to 5 = highly satisfied Do the various consumer segments (low, medium, and heavy users) differ in terms of their volume of product consumption? Product consumption measured using a ratio scale in # of Liters consumed Do the brand evaluations of consumers exposed to different commercials (groups that saw commercials #1, #2, and #3) vary? Brand evaluation measured using an interval scale from 1 = very negative to 7 = very positive

One-way ANOVA In an analysis of variance, we test whether the observed mean values for the same variable are different among three or more groups For example, we can test whether product sales are significantly different among retail stores distinguished by the level of in-store promotion they have in place. Null hypothesis is the hypothesis of no difference No, there is no significant difference in sales based on level of in-store promotion. Alternative hypothesis is the statement of a difference Yes, there is a significant difference in sales based on level of in-store promotion. An F-statistic (F) is used to test the significance of the difference.

Steps for conducting an ANOVA Write out the null and alternative hypotheses Calculate the F-statistic Use the significance level corresponding to the F statistic to determine whether or not there is support for a significant difference between at least two of the groups If the significance level is <.05 then you can reject the null hypothesis and conclude that there is a significant difference among the groups If the significance level is >.05 then you are unable to reject the null hypothesis and conclude that there is no significant difference among the groups in terms of the observed means Interpret the difference (or no difference) by discussing the observed means for the groups Display a means plot to help you interpret the results

SPSS EXAMPLE: One-way ANOVA RQ: Does level of in-store promotion have a significant effect on average retail store sales? Ho: Average sales do not differ based on level of in-store promotion. H1: Average sales are significantly influenced by level of in-store promotion Store (Store ID #) Promotion (ordinal scale) 1 = low level of in-store promotion 2 = medium level of in-store promotion 3 = high level of in-store promotion Sales (ratio scale) # of product units sold

SPSS EXAMPLE: One-way ANOVA Select ANALYZE Click on COMPARE MEANS Select ONE-WAY ANOVA Choose the independent variable “Promotion” Choose the dependent variable “Sales” Click on OPTIONS Select “Descriptive statistics” Select “Means plot” Click on CONTINUE Click on OK

SPSS EXAMPLE: One-way ANOVA .001<.05 Is there a significant difference in sales among stores based on level of promotion??? The F-statistic is equal to 15.00 with a corresponding significance level of .001. Since this significance level is <.05, we can reject the null hypothesis that there is no significant difference between the mean values of sales across the groups and conclude that the mean sales levels differ among groups of stores based on level of in-store promotion. Specifically, we can see that the highest mean number of products sold, equal to 9, occurred in the high level of in-store promotion group (n=5 stores). Thus, we can conclude that level of in-store promotion is positively related to sales.

SPSS EXAMPLE: One-way ANOVA NOTE: You can go back and conduct independent-samples t-tests to test for significant differences between 2 pairs of groups

Math: One-way ANOVA Step 1: State the null and alternative hypotheses Step 2: Assume the null hypothesis is true Step 3: Compute a relevant test statistic Step 4: Degrees of freedom df1 (numerator) = c-1; df2 (denominator) =N-c where c=# of groups & N=sample size Step 5: Compare observed F to a F-table with numerator and denominator dfs and appropriate significance level (.05)

What Hypothesis Tests Will We Learn? Test associations between two discrete variables Crosstabs and Chi-Squared test Test observed values of means or proportions of a single variable against a standard One sample t-test and One sample z-test Test observed values of means across two variables Paired samples t-test Test differences across two or more groups of respondents Independent samples t-test and ANOVA Test associations between two scaled variables Correlation Test the significance of regression coefficients that correspond to predictors in a regression model Regression