One way ANOVA

Question: Discuss about the various things for the analysis of responses differently given by the respondent regarding four categories of ANOVA? Answer: Introduction: We have to consider various things for the analysis of responses differently given by the respondent regarding four categories. Another thing to check is about average proportion, whether it is same for all five responses or not. Analysis also to take about responses whether respondent provides same responses to all type of question or not. We are dividing given questions into four categories that are legal, ethical, economic and philanthropic also five responses are considered such as strongly agree, agree, disagree, strongly disagree or impartial. Appropriate statistical test to perform for checking given hypothesis. So, we have to use ANOVA or one way analysis of variance. Following are the null hypothesis for one way ANOVA for this test: Null hypothesis: H0: Average proportions for all responses from respondent are same. Following is the alternative hypothesis for this test: Alternative hypothesis: Ha: Average proportions for all responses are not same. Following is data for this test: Data: 1. Strongly agree 2. Agree 3. Impartial 4. Disagree 5. Strongly disagree Economic 41 44 13 1 1 20.2 46.46 28.28 3.03 2.02 Legal 56.67 27.78 12.22 1.11 2.22 35.96 38.2 14.61 8.99 2.25 Ethical 21.84 47.13 22.99 5.75 2.3 12.64 24.14 29.89 28.74 4.6 Philanthropic 34.52 47.62 14.29 1.19 2.38 23.53 38.82 27.06 8.24 2.35 From given data in every category, two questions were asked and responses were recorded in percentage for every categories such as strongly agree, agree, impartial, disagree and strongly disagree. Following is the one way ANOVA test: One way ANOVA: Descriptive statistics for five responses is given in the following table: Descriptive Statistics N Minimum Maximum Mean Std. Deviation Strongly agree 8 12.64 56.67 30.7950 14.08667 Agree 8 24.14 47.62 39.2688 9.00679 Impartial 8 12.22 29.89 20.2925 7.51811 Disagree 8 1.00 28.74 7.2563 9.25335 Strongly disagree 8 1.00 4.60 2.3900 1.00081 Valid N (list wise) 8 The ANOVA table for this test is given as below: Analysis of Variance Source DF SS MS F P Factor 4 7683.3 1920.8 22.72 0.000 Error 35 2958.9 84.5 Total 39 10642.3 For this ANOVA test, we have p-value as 0.000.We take the level of significance for this test as 5% or alpha =0.05. Following are the decision rules: Reject null hypothesis when p-value alpha value Do not reject null hypothesis when p-value alpha value We have p-value alpha value So, from this we reject null hypothesis that average proportions for all responses from respondent are same. Interpretation: The result we interpret is that the Average proportions for all responses are not same . Reference: David R. Cox, D. V. Hinkley, Theoretical Statistics, Chapman Hall/CRC, 1979. Peter J. Bickel, Kjell A. Doksum, Mathematical Statistics, Volume 1, Basic Ideas and Selected Topics, 2rd ed. Prentice Hall, 2001. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic Press, Inc., New York, 1967 Harald Cramr, Mathematical Methods of Statistics, Princeton, 1946 Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer