2004/10/07Faculty Participation in CAT...1 Faculty Participation in the Center for the Advancement of Teaching, Xavier University of Louisiana New Orleans, LA Mathematics / Statistics Colloquium 2004 October 7 An Application of the Chi-square Probability Distribution V. J. DuRapau, Jr., Ph.D. Professor, Mathematics Department Todd Stanislav, Ph.D. Assoc. Prof., Biology Department Dir., Center for the Advancement of Teaching

Slide #’s are in the lower right-hand corner

This study was reviewed and approved by the chair of the Internal Review Board ( IRB ) of Xavier University of Louisiana. “The committee reviews research proposals which involve the use of human subjects. Any faculty member at Xavier who is using human subjects, or any research proposal which involves Xavier and uses human subjects must have the approval of the Xavier IRB. Use of human subjects includes tissues derived from them, such as skin, blood, organ, etc. Most surveys administered to students require IRB approval (except those conducted in the classroom as part of the educational process or those involving observation of public behavior).” ( Faculty Handbook, August 2003)

Acknowledgement Thanks to the staff of Xavier’s Office of Institutional Research for their help in compiling some of the data, namely the race/ethnicity data.

2004/10/07Faculty Participation in CAT...5 Contents n Introduction n Study Design and Procedures n Descriptive Statistics n Statistical Methodology n Results and Conclusions n Discussion n References

2004/10/07Faculty Participation in CAT...6 Introduction n Center for the Advancement of Teaching ä coordinates faculty development initiatives ä is an interdisciplinary, collaborative academic unit that seeks to focus the University's efforts aimed at advancing the art of teaching at all levels ä creates opportunities for Xavier faculty to develop new teaching strategies and to incorporate the use of technology in educationally effective ways ä supports Xavier faculty collaboration with preK-12 schools or teachers

2004/10/07Faculty Participation in CAT...7 Introduction (cont.) n Purpose of this study is to investigate several questions about ä proposals submitted to the Center ä proposals that were funded ä proposals that were not funded. n Is the gender distribution of faculty who submitted proposals to the Center during the same as the gender distribution of all full-time university faculty?

2004/10/07Faculty Participation in CAT...8 Introduction (cont.) n Is the race/ethnicity distribution of faculty who submitted proposals to the Center during approximately the same as the race/ethnicity distribution of all full-time university faculty? n Is the distribution of faculty by division for those who submitted proposals to the Center during approximately the same as the distribution of faculty by division for all faculty? n Is the distribution of faculty by department for those who submitted proposals to the Center during approximately the same as the distribution of faculty by department for all faculty?

2004/10/07Faculty Participation in CAT...9 Introduction (cont.) n Similar questions for ä Proposals that were funded ä Proposals that were not funded

2004/10/07Faculty Participation in CAT...10 Study Design and Procedures n Data collection n Data verification n Coding data

2004/10/07Faculty Participation in CAT...11 Data Collection n CAT staff generate Excel spreadsheet n For each faculty who submitted proposals during ä Faculty name ä Gender ä Race/ethnicity ä Division or department ä List of proposals submitted. For each, –Other faculty participating in the proposal –Identify proposals funded & those not funded

2004/10/07Faculty Participation in CAT...12 Data Verification n An essential step in any statistical analysis n 146 faculty submitted proposals (individual or joint proposals) ä Total of 251 proposals –209 funded –42 not funded n Generate a Lotus spreadsheet of aggregate information n Input data into SPSS (statistical program)

2004/10/07Faculty Participation in CAT...13 Coding Data – 10 cases (faculty)

2004/10/07Faculty Participation in CAT...14 Descriptive Statistics n XU Faculty: University Profile ä Why use the faculty data? –Small changes in distributions –See Appendix A of the full report for details n Faculty submitting proposals to CAT during

2004/10/07Faculty Participation in CAT...15 XU Faculty

2004/10/07Faculty Participation in CAT...16 XU Faculty: Division/Department Used FTE for Division and Department analyses

2004/10/07Faculty Participation in CAT...17 Statistical Methodology n Chi-square (χ 2 ) goodness of fit ä A hypothetical example ä Thousands of jellybeans! n Hypotheses tested

2004/10/07Faculty Participation in CAT...18 Chi-square Goodness of Fit n Consider a barrel of tens of thousands of jellybeans of k = 6 different colors n Suppose the proportions of each color are P i, i = 1, 2, …, 6 n If we were to draw a random sample of size n, we expect to have n. P i of color i n Now, draw a single random sample of n jellybeans n Let n i, i = 1, …, 6, denote the number of jellybeans in the sample of color i

2004/10/07Faculty Participation in CAT...19 Chi-square Goodness of Fit (cont) n Compute the following statistic: n Replace all the jellybeans in the barrel n Repeat this process of randomly selecting n jellybeans, noting the colors, and computing the χ 2 statistic

2004/10/07Faculty Participation in CAT...20 Chi-square Goodness of Fit (cont) n The result of this repeated sampling process is a large set of computed χ 2 statistics. n The distribution of these statistics is approximately the χ 2 distribution with k – 1 = 5 degrees of freedom. n The theory of the Chi-square goodness of fit test is based on the notion of repeated sampling. n In practice, we draw one sample and compare the computed χ 2 statistic against the theoretical distribution.

2004/10/07Faculty Participation in CAT...21 Chi-square Density Function The parameter λ is called the degrees of freedom (df).

2004/10/07Faculty Participation in CAT...22 Chi-square Density Function n Characteristics of any continuous probability density function (pdf) n Chi-square pdf is skewed to the right n Chi-square pdf is a single parameter function ä Degrees of freedom (λ) ä Mean of the distribution (λ) ä Variance of the distribution (2 λ)

2004/10/07Faculty Participation in CAT...23 χ 2 Probability Density Functions (pdf’s) Reference: Scientific WorkPlace, Computing Techniques (2003)

2004/10/07Faculty Participation in CAT...24 Chi-square pdf for df = 5 Significance level α = 0.05 From the jellybean example

2004/10/07Faculty Participation in CAT...25 Notation Used in Hypotheses n p represents the proportion of faculty submitting proposals (sample proportions) n P represents the proportion of all XU faculty (population proportions)

2004/10/07Faculty Participation in CAT...26 Hypotheses: Gender n Null hypothesis ä The gender distribution of faculty submitting proposals to CAT is the same as the gender distribution of all XU faculty. ä H 0 : p male = P male and p female = P female n Alternative hypothesis ä The gender distribution of faculty submitting proposals to CAT is different from the gender distribution of all XU faculty. ä H a : p male ≠ P male or p female ≠ P female

2004/10/07Faculty Participation in CAT...27 Hypotheses: Race/Ethnicity n Null hypothesis ä The race/ethnicity distribution of faculty submitting proposals to CAT is the same as the race/ethnicity distribution of all XU faculty. ä H 0 : p black = P black and p white = P white and p other = P other n Alternative hypothesis ä The race/ethnicity distribution of faculty submitting proposals to CAT is different from the race/ethnicity distribution of all XU faculty. ä H a : p black ≠ P black or p white ≠ P white or p other ≠ P other

2004/10/07Faculty Participation in CAT...28 Hypotheses: Division n Null hypothesis ä The distribution by division of faculty submitting proposals to CAT is the same as the distribution by division of all XU faculty. n Alternative hypothesis ä The distribution by division of faculty submitting proposals to CAT is different from the distribution by division of all XU faculty.

2004/10/07Faculty Participation in CAT...29 Hypotheses: Department n Null hypothesis ä The distribution by department of faculty submitting proposals to CAT is the same as the distribution by department of all XU faculty. n Alternative hypothesis ä The distribution by department of faculty submitting proposals to CAT is different from the distribution by department of all XU faculty.

2004/10/07Faculty Participation in CAT...30 Results and Conclusions n Gender n Race/ethnicity n Division n Department n By number of faculty n By proposals submitted n By proposals funded n By proposals not funded 16 separate Chi-square tests

2004/10/07Faculty Participation in CAT...31 Results: Gender (part 1) Differences not statistically significant

2004/10/07Faculty Participation in CAT...32 Results: Gender (part 2) Differences not statistically significant

2004/10/07Faculty Participation in CAT...33 Results: Race/Ethnicity (Part 1) Differences not statistically significant

2004/10/07Faculty Participation in CAT...34 Results: Race/Ethnicity (Part 2) Differences not statistically significant

2004/10/07Faculty Participation in CAT...35 Results: Division (Part 1) Differences are statistically significant

2004/10/07Faculty Participation in CAT...36 Results: Division (Part 2) Differences statistically sig. Differences not statistically sig.

2004/10/07Faculty Participation in CAT...37 Results: Department (Part 1)

2004/10/07Faculty Participation in CAT...38 Results: Department (Part 2)

2004/10/07Faculty Participation in CAT...39 Discussion n Gender and race/ethnicity distributions of faculty participating in funded activities of the CAT matches the corresponding distributions of full-time faculty. n There are some differences with respect to division and with respect to department. n Asymptotic significances (p-values) were used in these analyses. (Similar to approximating a binomial distribution with a normal distribution.) ä Analyses using exact p-values might yield additional insight. n Multiple significance testing (16 in this report) increase the risk of Type 1 errors (rejecting a true null hypothesis). n Alternative analyses (e.g., logistic regression models) should be considered.

2004/10/07Faculty Participation in CAT...40 References n Center for the Advancement of Teaching web site (2004). Xavier University of Louisiana, Mission Statement, n Hawkes, J. & Marsh, W. (2005). Discovering Statistics (2nd ed). Hawkes Learning Systems and Quant Systems, Inc.: Charlotte, NC. n Ott, R. (1993). An Introduction to Statistical Methods and Data Analysis (4th ed). Duxbury Press: Pacific Grove, CA. n Scientific WorkPlace 5.0 (2003) MacKichan Software Inc. ( n Wackerly, D., Mendenhall, W., & Scheaffer, R. (2002). Mathematical Statistics with Applications (6th ed). Duxbury Press: Pacific Grove, CA. n Xavier University Profiles. Office of Institutional Research.