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CHAPTER fifteen Learning Objectives Statistical Testing of Differences Copyright © 2002 South-Western/Thomson Learning Learning Objectives Learning Objectives 1. To become aware of the nature of statistical significance. 2. To understand the concept of hypothesis development and how to test hypotheses. 3. To understand the differences between Type I and Type II errors. 4. To be familiar with several of the more common statistical tests of goodness of fit, hypotheses about one mean, hypothesis about two means, and hypotheses about proportions. 5. To learn about analysis of variance. Learning Objectives Differences and Changes To become aware of the nature of statistical differences. Are certain measures different from one another? For example: Did top-of-mind awareness really increase? Did customer satisfaction really increase? Learning Objectives Statistical Significance To become aware of the nature of statistical differences. It is possible for numbers to be different in a mathematical sense but not statistically different in a statistical sense. • Mathematical differences • Statistical significance • Managerially important differences Hypothesis Testing Learning Objectives To understand the concept of hypothesis development and how to test hypotheses. Hypothesis An assumption that a researcher makes about some characteristic of the population under study. Steps in Hypothesis Testing Step One: Stating the Hypothesis Null hypothesis: Ho Alternative hypothesis: Ha Step Two: Choosing the Appropriate Test Statistic Hypothesis Testing Learning Objectives To understand the concept of hypothesis development and how to test hypotheses. Step Three: Developing a Decision Rule Step Four: Calculating the Value of the Test Statistic • Use the appropriate formula • Compare calculated value to the critical value. • State the result in terms of: • rejecting the null hypothesis • failing to reject the null hypothesis Step Five: Stating the Conclusion Other issues Learning Objectives To understand the differences between Type I and Type II errors. Types of Errors in Hypothesis Testing Type I Error Rejection of the null hypothesis when, in fact, it is true. Type II Error Acceptance of the null hypothesis when, in fact, it is false. Accepting Ho or Failing to Reject Ho? One-Tailed Test or Two-Tailed Test? Table 15.2 Table 15.2 Learning Objectives Type I and Type II Errors Actual State of the Null Hypothesis Fail to Reject Ho Reject Ho Ho is true Correct (1-) no error Type I error () Ho is false Type II error () Correct (1- ) no error Commonly Used Statistical Hypothesis Tests Learning Objectives To understand the concept of hypothesis development and testing a hypothesis. Independent Versus Related Samples Independent samples Measurement of a variable in one population has no effect on the measurement of the other variable Related Samples Measurement of a variable in one population may influence the measurement of the other variable. Degrees of Freedom The number of observations minus the number of constraints. Learning Objectives Goodness of Fit To describe statistical tests of goodness of fit. Chi-Square To determine whether an observed pattern of frequencies corresponds to an “expected” pattern. Chi-Square Test of a Single Sample 1. Specify the null and alternative hypotheses. 2. Determine the number of visitors that would be expected in each category if the null hypotheses were correct (Ei). Learning Objectives Goodness of Fit To describe statistical tests of goodness of fit. 3. Calculate the X2 value using the formula: X2 = k (Oi - Ei ) 2 I=1 Ei Oi = observed number in i th category Ei = expected number in i th category k = number of categories Learning Objectives Goodness of Fit To describe statistical tests of goodness of fit. 4. Select the level of significance . 5. Compare the calculated X2 value with the table value. State the result. Chi-Square Test of Two Independent Samples Association between two or more variables. 1. State the null and alternative hypotheses. Learning Objectives Goodness of Fit To describe statistical tests of goodness of fit. 2. Place the observed (sample) frequencies in a k x r table using the k columns for the sample groups and the r rows for the conditions or treatments. Calculate the sum of each row and each column. Record marginal totals. Calculate total for entire table. 3. Determine the expected frequency for each cell in the contingency table by calculating the product of the two marginal totals common to that cell and dividing that value by N. Learning Objectives Goodness of Fit To describe statistical tests of goodness of fit. 4. Calculate the value of X2 using: X2 = r k I=1 j=1 (Oij - E ij) 2 Eij Oij = observed number in i th row of the jth column Eij = expected number in i th row of the jth column 5. Compare the calculated X2 with the tabular X2. State the result. Learning Objectives Goodness of Fit To describe statistical tests of goodness of fit. Kolmogorov-Smirnov Test 1. Specify the null and alternative hypotheses. 2. Establish the cumulative frequency distribution expected under the null hypothesis. 3. Calculate the observed cumulative frequency distribution from the sample. 4. Select the level of significance . 5. Determine the K-S statistic. 6. State the result. Learning Objectives About One Mean To describe statistical tests about one mean. Z - Test Hypothesis tests about a single mean if the sample mean if the sample is large enough and drawn from a normal population. 1. Specify the null and alternative hypotheses. 2. Specify the level of sampling error () allowed. 3. Determine the sample standard deviation. 4. Calculate the estimated standard error of the mean: Learning Objectives About One Mean Sx = To describe statistical tests about one mean. S √n where Sx = the estimated standard error of the mean. Learning Objectives About One Mean To describe statistical tests about one mean. 5. Calculate the test statistic: Z= (sample mean) population mean under the null hypothesis estimated standard error of the mean 6. State the result. Learning Objectives About One Mean To describe statistical tests about one mean. t - Test Hypothesis tests about a single mean if the sample mean if the sample is too small to use the Z - test 1. Specify the null and alternative hypotheses. 2. Specify the level of sampling error () allowed. 3. Determine the sample standard deviation. Learning Objectives To describe statistical tests about one mean. About One Mean S = √ n (Xi - X) 2 I=1 n-1 where Xi = observed sales per week in the ith store X = average sales per week n = the number of stores Learning Objectives About One Mean To describe statistical tests about one mean. 4. Calculate the estimated standard error of the mean using: Sx = S √n Learning Objectives About One Mean To describe statistical tests about one mean. 5. Calculate the t - statistic: (sample mean) t= population mean under the null hypothesis estimated standard error of the mean 6. State the result. Learning Objectives About Two Means To describe statistical tests about two means. Marketers are frequently interested in testing differences between groups. 1. The null and alternative hypotheses 2. Set the level of sampling error () 3. Estimate the standard error of the differences between two means: Learning Objectives To describe statistical tests about two means. About Two Means Sx m-ƒ = √ S2m nm + S2 ƒ nƒ Sm = estimated standard deviation of population m (men) Sf = estimated standard deviation of population f (women) nm = sample size for sample m nf = sample size for sample f Learning Objectives About Two Means To describe statistical tests about two means. 4. Calculate the test statistic. Z is calculated as: difference between means of first sample and second sample Z= difference between means under the null hypothesis standard error or the differences between the means 5. Compare the calculated value of Z with the critical value. State the result. Learning Objectives About Proportions To describe statistical tests about proportions. Test of a Proportion, One Sample Phenomena expressed as percentages 1. Specify the null and alternative hypotheses 2. Set the level of sampling error () 3. Estimate the standard error using the P-value specified in the null hypothesis: Learning Objectives About Proportions Sp = To describe statistical tests about proportions. √ P(1- P) n-1 where: P = proportion specified in the null hypothesis n = sample size Learning Objectives About Proportions To describe statistical tests about proportions. 4. Calculate the test statistic as follows: Z= (observed proportion - proportion under null hypothesis) estimated standard error (Sp) Compare the calculated Z-value with the critical Z-value. Learning Objectives About Proportions To describe statistical tests about proportions. Tests of Differences Between Two Proportions, Independent Samples Management may be interested in the difference between the proportions of people in two different groups that engage in a certain activity or have a certain characteristic. 1. Specify the null and alternative hypotheses 2. Set the level of sampling error () 3. Estimate the standard error of the differences between the two proportions: Learning Objectives About Proportions Spm-ƒ where P= = To describe statistical tests about proportions. √ P(1- P) 1 nm nmPm + nfPf nm + nf Pm = proportion in sample m (men) Pf = proportion in sample f (women) nm = sample of size m nf = sample of size f + 1 nƒ Learning Objectives About Proportions To describe statistical tests about proportions. 4. Calculate the test statistic: difference between observed proportions Z= difference between proportions under the null hypothesis estimated standard error or the differences between the two means 5. Compare the calculated Z-value with the critical Z-value. State the result. Learning Objectives To learn about analysis of variance. Analysis of Variance Analysis of Variance (ANOVA) Test for the differences among the means of two or more variables. 1. Specify the null and alternative hypotheses 2. Calculate sum of squares among groups (SSA) SSA = j= 1 nj (X - X ) 2 i t Learning Objectives Analysis of Variance To learn about analysis of variance. 3. Calculate the variation among the group means as measured by the mean sum of squares among groups (MSA): MSA = sum of squares among groups (SSA) degrees of freedom (d.f.) where Degrees of freedom = number of groups (C) -1 Learning Objectives To learn about analysis of variance. Analysis of Variance 4. Calculate the sum of squares within groups or sum of squared error (SSE): nj c SSE = j=1 I=1 (Xij- Xj) 2 5. Calculate the mean square error (MSE): MSE = sum of squares within groups (SSE) degrees of freedom (d.f.) Learning Objectives Analysis of Variance To learn about analysis of variance. 6. Calculate the F - statistic as follows: F= MSA MSE 7. State the result. p - VALUES AND SIGNIFICANCE TESTING 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. Learning Objectives SUMMARY • Differences and Changes • Statistical Significance • Hypothesis Testing • Other issues • Hypothesis Tests • Goodness of Fit • About One Mean • About Two Means • About Proportions • Analysis of Variance Learning Objectives The End Copyright © 2002 South-Western/Thomson Learning