Intuitive Introductory Statistics pp 331-415 | Cite as

# Sampling Distributions and Approximations

## Abstract

Now that we have built a solid foundation based on exploratory data analysis techniques, proper design of experiments, and basic probability, we will put the finishing touches on this foundation by studying sampling distributions, the most important part of statistical inference. Before we can use the information we have collected and analyzed from our sample to make inferences about the population of interest, we must make sure that we understand how the statistic we have computed varies with repeated sampling. Our goal here is to describe what might happen if we repeat the entire sampling process and computation of the desired statistic again and again. Do you think that if you take a different sample you will get exactly the same value for the statistic? While it is certainly possible that this could happen, in most practical settings it is very unlikely that you will get exactly the same value of the statistic. At first, it might appear that this would be a major problem for the field of statistics. If we collect different samples and they usually give us different results, how can we make any inferences? The fact is that even though the values of the statistic are likely to differ from sample to sample, they will follow a pattern. This pattern of variation in repeated sampling is described by the sampling distribution of the statistic.

## Supplementary material

## Bibliography

- Ault, R. G., Hudson, E. J., Linehan, D. J., & Woodward, J. D. (1967). A practical approach to the assessment of head retention of bottled beers.
*Journal of the Institute of Brewing, 73*(6), 558–566.CrossRefGoogle Scholar - Baxley, F., & Miller, M. (2006). Parental misperceptions about children and firearms.
*Archives of Pediatric & Adolescent Medicine, 160*(5), 542–547.CrossRefGoogle Scholar - Devanand, D. P., Lee, S., Manly, J., Andrews, H., Schupf, N., Masurkar, A., Stern, Y., Mayeux, R., & Doty, R. L. (2015). Oflactory identification deficits and increased mortality in the community.
*Annals of Neurology, 78*(3), 401–411.CrossRefGoogle Scholar - Gallup, Inc. (2012a).
*Business Journal: Your employees don’t “Get” your brand. J. H. Fleming, and D. Witters.*Report issued July 26, 2012. www.gallup.com - Gallup, Inc. (2012b).
*Economy: U. S. workers least happy with their work stress and pay.*Report issued November 12, 2012. www.gallup.com - Gallup, Inc. (2014).
*Well-being: Americans favor ban on smoking in public, but not total ban.*R. Riffkin; report issued July 30, 2014. www.gallup.com - Gallup, Inc. (2015d).
*Politics: More republicans now prefer one-party Government*. J. M. Jones; report issued October 12, 2015. www.gallup.com - Intellicast. (2016). http://www.intellicast.com/Local/History.aspx?month=9. Accessed 30 June 2016.
- Mackowiak, P. A., Wasserman, S. S., & Levine, M. M. (1992). A critical appraisal of 98.6
^{o}*F*, the upper limit of the normal body temperature, and other legacies of Carl Reinhold August Wunderlich.*Journal of the American Medical Association, 268*(12), 1578–1580.CrossRefGoogle Scholar - Moore, T. L. (2006). Paradoxes in film ratings.
*Journal of Statistics Education, 14*(1), 8 pages online.Google Scholar - National Public Radio. (2014). Scott Neuman, February 14, 2014. http://www.npr.org/sections/thetwo-way/2014/02/14/277058739/. Accessed 21 July 2016.
- National Pubic Radio. (2015).
*Sports and health in America. Prepared in conjunction with the Robert Wood Johnson Foundation and the Harvard T. H. Chan School of Public Health*. Released June 2015. www.npr.org - Neuman, M. D., & Werner, R. M. (2015). Marital status and postoperative functional recovery.
*Journal of the American Medical Association Surgery, 151*, 194, Published online October 28, 2015, 3 pages.Google Scholar - Shkedy, Z., Aerts, M., & Callaert, H. (2006). The weight of Euro coins: Its distribution might not be as normal as you would expect.
*Journal of Statistics Education, 14*(2), 15 pages online.Google Scholar - Shoemaker, A. L. (1996). What’s normal?—Temperature, gender, and heart rate.
*Journal of Statistics Education, 4*(2), 4 pages online.Google Scholar - USA Today. (2013).
*Chris Chase, USA TODAY Sports*. January 30, 2013. http://www.usatoday.com/story/gameon/2013/01/30. Accessed 21 June 2016. - VisitingAngels. (2013).
*National survey reveals children choose Mom over Dad*. Released June 7, 2013. www.visitingangels.com - Wolfe, J., Martinez, R., & Scott, W. A. (1998). Baseball and beer: An analysis of alcohol consumption patterns among male spectators at major-league sporting events.
*Annals of Emergency Medicine, 31*, 629–632.CrossRefGoogle Scholar - Woodward, W. F. (1970).
*A comparison of base running methods in baseball*. MSc thesis, Florida State University.Google Scholar - Wypijewski, J. (1997).
*Painting by numbers: Komar and Melamid’s*s*cientific guide to art.*Farrar, Straus, & Giroux, Inc.Google Scholar