Interval estimationmethodological approaches and understanding difficulties

  1. Batanero, Carmen 1
  2. Díaz-Batanero, Carmen 2
  3. López-Martín, María del Mar 3
  4. Roldán López de Hierro, Antonio Francisco 1
  1. 1 Universidad de Granada
    info

    Universidad de Granada

    Granada, España

    ROR https://ror.org/04njjy449

  2. 2 Universidad de Huelva
    info

    Universidad de Huelva

    Huelva, España

    ROR https://ror.org/03a1kt624

  3. 3 Universidad de Almería
    info

    Universidad de Almería

    Almería, España

    ROR https://ror.org/003d3xx08

Revista:
BEIO, Boletín de Estadística e Investigación Operativa

ISSN: 1889-3805

Año de publicación: 2020

Volumen: 36

Número: 3

Páginas: 269-291

Tipo: Artículo

Otras publicaciones en: BEIO, Boletín de Estadística e Investigación Operativa

Resumen

Despite the wide use of interval estimation to complement or replace hypothesis tests, didactic research has described widespread errors in its understanding or application. Although the interval estimation can be carried out from several methodological approaches, the current teaching focuses almost exclusively on the frequentist methodology. This paper begins by describing the main approaches of the interval estimation and summarizes the most common mistakes in their interpretation. The aim is helping statistics teachers to diagnose the difficulties of their students in this subject. Finally, we include some considerations on teaching and some informal approaches.

Referencias bibliográficas

  • [1] Batanero, C. y Borovcnik, M. (2016). Statistics and probability in high school. Sense Publishers, Rotterdam (The Netherlands).
  • [2] Bayes, T. (1970). An essay towards solving a problem in the doctrine of chances. En E. S. Pearson y M. G. Kendall (Eds.), Studies in the history of statistics and probability (Vol. 1, pp. 131-153). Londres: Griffin (trabajo original publicado en 1763).
  • [3] Begu´e, N., Batanero, C., Ruiz, K. y Gea, M.M. (2019). Understanding sampling: a summary of the research. Bolet´ın de Estad´ıstica e Investigaci´on Operativa, 35(1), 49-78.
  • [4] Behar, R. (2001). Aportaciones para la mejora del proceso de ense˜nanzaaprendizaje de la estad´ıstica. Tesis Doctoral. Universidad Polit´ecnica de Catalu˜na.
  • [5] Belia, S., Fidler, F. y Cumming, G. (2005). Researchers misunderstand confidence intervals and standar error bars. Psychological Methods, 4, 389- 396.
  • [6] Ben-Zvi, D., Bakker, A. y Makar, K. (2015). Learning to reason from samples. Educational Studies in Mathematics, 88(3), 291-303.
  • [7] Bolstad, W. (2013). Introduction to Bayesian statistics (2.a ed.). Wiley, Nueva York (EEUU).
  • [8] Borovcnik, M. (2019). Informal and “informal” inference. En J. M. Contreras, M. M. Gea, M. M. L´opez-Mart´ın y E. Molina-Portillo (Eds.), Actas del Tercer Congreso Internacional Virtual de Educaci´on Estad´ıstica. Disponible en www.ugr.es/local/fqm126/civeest.html
  • [9] Cabri´a, S. (1994). Filosof´ıa de la estad´ıstica. Servicio de Publicaciones de la Universidad, Valencia (Espa˜na).
  • [10] Castro Sotos, A. E., Vanhoof, S., Van den Nororgate, W. y Onghena, P. (2007). Student’s misconceptions of statistical inference: A review of the empirical evidence form research on statistical education. Educational Research Review, 2(2), 98-113.
  • [11] Cepeda-Cuervo, E., Aguilar, W., Cervantes, V., Corrales, M., D´ıaz, I. y Rodr´ıguez, D. (2008). Intervalos de confianza e intervalos de credibilidad para una proporci´on. Revista Colombiana de Estad´ıstica, 31(2), 211-228.
  • [12] Cobb, G. W. (2007). The introductory statistics course: A Ptolemaic curriculum. Technology Innovations in Statistics Education, 1(1), 1-15.
  • [13] Cumming, G. y Finch, S. (2005). Inference by eye: confidence intervals and how to read pictures of data. American Psychologist, 60(2), 170.
  • [14] Cumming, G., Williams, J. y Fidler, F. (2004). Replication, and researchers’ understanding of confidence intervals and standard error bars. Understanding Statistics, 18(3), 299-311. doi: 10.1111/j.1467-9280.2007.01881.x
  • [15] De la Fuente, E. I. y D´ıaz-Batanero, C. (2004). Controversias en el uso de la inferencia en la investigaci´on experimental. Metodolog´ıa de las Ciencias del Comportamiento, Volumen especial 2004, 161-167.
  • [16] D´ıaz Batanero, C. (2007). Viabilidad de la ense˜nanza de la inferencia bayesiana en el an´alisis de datos en psicolog´ıa. Tesis doctoral. Universidad de Granada.
  • [17] D´ıaz-Batanero, C. (2018). Proyecto docente. Huelva: La autora.
  • [18] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics, 7, 1-26.
  • [19] Efron, B. y Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals and other measures of statistical accuracy. Statistical Sience, 1, 54-75.
  • [20] Engel, J. (2010). On teaching bootstrap confidence intervals. In C. Reading (Ed.), Proceedings of the Ninth International Conference on Teaching of Statistics. Voorburg: International Statistical Institute.
  • [21] Estes, W. K. (1997). On the communication of information by displays of standard errors and confidence intervals. Psychonomic Bulletin & Review, 4(3), 330-341.
  • [22] Fidler, F. y Cumming, G. (2005). Teaching confidence intervals: Problems and potential solutions. Proceedings of the 55th International Statistics Institute, Session CD-ROM. Sidney, Australia: International Statistical Institute.
  • [23] Gelman, A. y Shalizi, C. R. (2012). Philosophy and the practice of Bayesian statistics. British Journal of Mathematical and Statistical Psychology, 66(1), 8-38.
  • [24] Gil-Flores, J. (2005). Aplicaci´on del m´etodo Bootstrap al contraste de hip´otesis en la investigaci´on educativa. Revista de Educaci´on, 336, 251- 265.
  • [25] Harradine, A., Batanero, C. y Rossman, A. (2011). Students and teachers’ knowledge of sampling and inference. En C. Batanero, G. Burrill y C. Reading (Eds.), Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education (pp. 235-246). Springer Netherlands.
  • [26] Hesterberg, T., Monaghan, S., Moore, D. S., Clipson, A., y Epstein, R. (2003). Bootstrap methods and permutation tests: En D. Moore, G. McCabe, W. Duckworth y L. Alwan (Eds.). The practice of business statistics (pp. 1-70). Nueva York.
  • [27] Howell, D. (n.d.). Resampling statistics: Randomization & Bootstrap. Statistical page Howell. Online: www.uvm.edu/\protect\char126\ relaxdhowell/StatPages/Resampling/Resampling.html
  • [28] Lecoutre, B. (2006). Training students and researchers in Bayesian methods for experimental data analysis. Journal of Data Science, 4(2), 207-232.
  • [29] Lecoutre, B., y Poitevineau, J. (2014). The significance test controversy revisited. Springer, Berlin, Heidelberg.
  • [30] L´opez-Mart´ın, M. M., Batanero, C., D´ıaz-Batanero, C. y Gea, M. M. (2016). La inferencia estad´ıstica en las Pruebas de Acceso a la Universidad en Andaluc´ıa. Revista Paranaense de Educa¸c˜ao Matem´atica, 5(8), 33-59.
  • [31] L´opez-Mart´ın, M. M., Batanero, C. y Gea, M. M. (2019a). Prospective high school teachers’ interpretation of hypothesis tests and confidence intervals. Trabajo presentado en CERME 11, Utrecht, Febrero, 2019.
  • [32] L´opez-Mart´ın, M. M., Batanero, C. y Gea, M. M. (2019b). ¿Conocen los futuros profesores los errores de sus estudiantes en inferencia? Bolema, 33(64), en prensa.
  • [33] Mayo, D. G. (1981). In defense of the Neyman-Pearson theory of statistics. Philosophy of Science, 48, 269-280.
  • [34] Mayo, D. G. y Cox, D. R. (2006). Frequentist statistics as a theory of inductive inference. IMS Lecture Notes-Monograph Series, 49, 77-97.
  • [35] Ministerio de Educaci´on, Cultura y Deporte, MECD (2015). Real Decreto 1105/2014, de 26 de diciembre, por el que se establece el curr´ıculo b´asico de la Educaci´on Secundaria Obligatoria y del Bachillerato. Madrid: Autor.
  • [36] Mooney, C. Z., Duval, R. D. y Duval, R. (1993). Bootstrapping: A nonparametric approach to statistical inference. Sage, Londres (Inglaterra).
  • [37] Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., y Wagenmakers, E. J. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic Bulletin & Review, 23(1), 103-123.
  • [38] Neyman, J. (1934). On the two different aspects of the representative method. Journal of the Royal Statistical Society, 97, 558-625.
  • [39] Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philosofical Transaction of the Royal Society of London, series A, Mathematical and Physical Sciences, 236(767), 33-380.
  • [40] Olivo, E. (2008). Significados del intervalo de confianza en la ense˜nanza de la ingenier´ıa en M´exico. Tesis Doctoral. Universidad de Granada.
  • [41] Olivo, E. y Batanero, C. (2007). Un estudio exploratorio de dificultades de comprensi´on del intervalo de confianza. Uni´on, 12, 37-51.
  • [42] Olivo, E., Batanero, C. y D´ıaz, C. (2008). Dificultades de comprensi´on del intervalo de confianza en estudiantes universitarios. Educaci´on Matem´atica, 20(3), 5-32.
  • [43] Parsonage, R., Pfannkuch, M., Wild, C. J., & Aloisio, K. (2016). Bootstrapping confidence intervals. In D. Ben-Zvi y K. Makar (Eds.), The teaching and learning of statistics (pp. 181-191). Springer, Cham.
  • [44] Pfannkuch, M., Wild, C. J. y Parsonage, R. (2012). A conceptual pathway to confidence intervals. ZDM, 44(7), 899-911.
  • [45] Quenouille, M. H. (1949). Approximate tests of correlation in time-series. Mathematical Proceedings of the Cambridge Philosophical Society, 45(3), 483-484.
  • [46] Ramos, C. E., Espinel, M.C., y Ramos, R. M. (2009). Identificaci´on de los errores en los contrastes de hip´otesis de los alumnos de Bachillerato. SUMA, 61, 35-44.
  • [47] Rivadulla, A. (1991). Probabilidad e inferencia cient´ıfica. Anthropos, Barcelona (Espa˜na).
  • [48] Rossman, A. J. (2008). Reasoning about Informal Statistical Inference: One Statistician’s View. Statistics Education Research Journal, 7(2), 5-19.
  • [49] Rouanet, H. (1998). Statistics for researchers. En H. Rouanet et al. (Eds.), New ways in statistical methodology (pp. 1 – 28). Berna: Peter Lang.
  • [50] Thompson, P., Liu, Y., y Saldanha, L. (2007). Intricacies of statistical inference and teachers’ understandings of them. In M. Lovett y P. Shaw (Eds.), Thinking with data (pp. 207–231). Mawah, NJ: Erlbaum.
  • [51] Tob´ıas-Lara, M. G. y G´omez-Blancarte, A. L. (2019). Assessment of informal and formal inferential reasoning: a critical research review. Statistics Education Research Journal. 18(1), 8-25.
  • [52] Tukey, J. W. (1956). Bias and confidence in non-quite large samples. Annals of Mathematical Statistics, 29, 614.
  • [53] Wilkinson, L. y Task Force on Statistical Inference (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604.
  • [54] Y´a˜nez, G. y Behar, R. (2009). Interpretaciones erradas del nivel de confianza en los intervalos de confianza y algunas explicaciones plausibles. En M.J. Gonz´alez, M.T. Gonz´alez y J. Murillo (Eds.). Investigaci´on en Educaci´on Matem´atica XIII, Santander: SEIEM.
  • [55] Yaremko, R. M., Harari, H., Harrison, R. C. y Lynn, E. (2013). Handbook of research and quantitative methods in psychology: For students and professionals. Hilldale, NJ: Erlbaum.