Chapter 15 Data Analysis: Testing for Significant Differences

Value of Testing for Data Differences Analysis of variance (ANOVA) Hypothesis testing t-distribution and associated confidence interval estimation Central tendency and dispersion Common to all marketing research projects Common to all marketing research projects Basic Statistics and Descriptive Analysis 15-2

Mode – most common value in a set of responses; i.e., the question response most often given. (Nominal data) Median – middle value of a rank ordered distribution; half of the responses are above and half below the median value. (Ordinal data) Mean – average of the sample. (Interval and Ratio data) Range – the distance between the smallest and largest values of the variable. Standard deviation – the average distance of the dispersion of the values from the mean. Variance – the squared standard deviation. Statistical Measures 15-3

Inferential statistics – to make a determination about a population on the basis of a sample. o Sample – a subset of the population. o Sample statistics – measures obtained directly from sample data. o Population parameter – a measured characteristic of the population. Actual population parameters are unknown since the cost to perform a census of the population is prohibitive. Analyzing Relationships of Sample Data 15-4

Type I: Rejection of the null hypothesis when, in fact, it is true. (Convict an innocent man) Type II: Acceptance of the null hypothesis when, in fact, it is false. (Set a guilty man free) Tests are either one or two-tailed. This approach depends on the nature of the situation and what the researcher is demonstrating. One-Tailed Tests (provide some direction) “If you take the medicine, you will get better” Two-Tailed Tests (there is a difference) “If you take the medicine, you will get either better or worse.” Chapter 15 Hypothesis Testing – Error Types

Chapter 15 Significance Testing Error Issues Type I and Type II Errors

Alpha and Beta (chances of making a Type I or Type II error) are related. If we set one very small (Alpha is.0001%), then we make the other very large. We are most concerned with minimizing Alpha, therefore a common percentage for a typical research study is 5% so we are 95% confident in the results. Chapter 15 Significance Testing Error Issues

When do we do a means test? When do we do a proportions test? When do we do a chi-square (frequency distribution) test? In making these decisions, we consider the level of data. We cannot calculate a mean on nominal or ordinal data, therefore we must do a proportion of frequency distribution test. If we have interval or ratio level data, we typically conduct a means test. If we have nominal or ordinal data, and we have one group or two groups, we usually do a proportion’s z-test. If we have more than two groups, we usually do a chi-square test. Chapter 15 Types of Hypothesis Tests

One Mean Used to test whether a sample mean is significantly different from an expected or pre-determined mean. Z-Test - usually for samples of about 30 and above. t-Test - usually for samples below 30. Two Means Z-Test Tests the difference between means for two. More than Two Means ANOVA (Analysis of Variance) Tests the difference between means for more than two groups. Chapter 15 Types of Mean Hypothesis Tests

Chapter 15 Types of Proportion Hypothesis Tests One Sample Z-Test Test to determine whether the difference between proportions is greater than would be expected because of sampling error. Two Proportions in Independent Samples Z-Test Test to determine the proportional differences between two or more groups. More than Two Groups Chi-square χ 2 Test to determine whether the difference between three or more groups is greater than would be expected.

p-value The exact probability of getting a computed test statistic that was largely due to chance. The smaller the p-value, the smaller the probability that the observed result occurred by chance. The smaller the p-value, the more likely the test is significant. For example a p-value of.045 is equivalent to a statistically significant test at a 95.5% level of confidence (4.5% alpha level). Chapter 15 Hypothesis Testing Term

Independent samples – two or more groups of respondents that are tested as though they may come from different populations (independent samples t-test). Related samples – two or more groups of respondents that originated from the sample population (paired samples t-test). Paired samples – questions are independent but respondents are the same. Hypothesis Testing 15-12

Analyzing Relationships of Sample Data Bivariate Hypothesis Bivariate Hypothesis Null Hypothesis Null Hypothesis... more than one group is involved.... there is no difference between the group means. µ1 = µ2 or that µ1 - µ2 = 0... there is no difference between the group means. µ1 = µ2 or that µ1 - µ2 =

Bivariate Statistical Tests Cross-tabulation – is useful for examining relationships and reporting the findings for two variables. The purpose of cross-tabulation is to determine if differences exist between subgroups of the total sample

Chi-Square (X 2 ) Analysis... test for significance between the frequency distributions of two or more nominally scaled variables in a cross-tabulation table to determine if there is any association

Assesses how closely the observed frequencies fit the pattern of the expected frequencies and is referred to as a ”goodness-of-fit” test. Used to analyze nominal data which cannot be analyzed with other types of statistical analysis, such as ANOVA or t-tests. Results will be distorted if more than 20 percent of the cells have an expected count of less than 5. Chi-Square (X 2 ) Analysis 15-16

Analyzing Data Relationships Requirements for ANOVA o dependent variable can be either interval or ratio scaled. o independent variable is categorical. Null hypothesis for ANOVA – states there is no difference between the groups – the null hypothesis is... o µ1 = µ2 = µ

ANOVA Total variance – separated into between-group and within-group variance. F-test – used to statistically evaluate the differences between the group means. Determining Statistical Significance Determining Statistical Significance 15-18

ANOVA – Testing Statistical Significance The larger the F ratio the larger the difference in the variance between groups.... the more likely the null hypothesis will be rejected. Based on the F- distribution Examines the ratio of two components of total variance and is calculated as shown below... F ratio = Variance between groups Variance within groups F ratio = Variance between groups Variance within groups... implies significant differences between the groups

Analyzing Relationships of Sample Data ANOVA – cannot identify which pairs of means are significantly different from each other. Must perform follow-up tests to identify the means that are statistically different from each other. Including: Sheffé Tukey, Duncan and Dunn 15-20

Analyzing Data Relationships ANOVA (analysis of variance) ANOVA (analysis of variance)... determines if three or more means are statistically different from each other (single dependent variable)... same as ANOVA but multiple dependent variables can be analyzed together. MANOVA (multivariate analysis of variance) MANOVA (multivariate analysis of variance) 15-21

Perceptual Maps... have a vertical and a horizontal axis that are labeled with descriptive adjectives. To develop perceptual maps – can use rankings, mean ratings, and multivariate methods

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Perceptual Mapping Distribution Advertising Image development New product development Applications in Marketing Research Applications in Marketing Research 15-24