invest_3ed.pdf

# Large large enough for the normal approximation to be

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large (large enough for the normal approximation to be valid), we saw that these confidence intervals have a very convenient form: statistic + margin-of-error where margin-of-error = critical value × standard error of statistic. The critical value is the number of standard errors you want to use corresponding to a specified confidence level. Keep in mind that the level of confidence provides a measure of how reliable the procedure will be in the long-run (which can vary by procedure and sample conditions). Finally, you saw that this reasoning process holds equally well when the sampling is from a finite population, where the randomness in our model comes from the selection of the observational units, not in the observational units’ individual outcomes. Here we took a bit more care to convince ourselves that the sample will be representative of the larger population. This is done by using random mechanisms to

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Chance/Rossman, 2015 ISCAM III Chapter 1 Summary 130 select the sample. These random mechanisms (e.g., simple random sampling) lead to samples that will generally have characteristics mirroring those of the larger population. Although random sampling prevents systematic sampling errors, you still need to worry about non-sampling errors (e.g., wording of a question). Also, there will still be random sampling variability the characteristics of the sample will vary from sample to sample. Technically we should use the hypergeometric distribution to model the behavior of the statistic. But we saw that if the population is large compared to the size of the sample (e.g., more than 20 times larger), then we can use the same binomial distribution and if the sample size is also large we can use normal-based methods to determine p-values and confidence intervals (as well as power and sample size calculations). The interpretation of the p-value is essentially the same but now applies to the randomness inherent in the sampling process. Also keep in mind that the confidence interval aims to capture the proportion of the population having the characteristic of interest. (Note, when the population is large, these are actually equivalent because the probability of any one randomly selected observational unit being a success will equal the population proportion of successes and we are approximating this as constant for every member of the sample.) SUMMARY OF WHAT YOU HAVE LEARNED IN THIS CHAPTER x The reasoning process of statistical inference x The terms parameter to describe a numerical characteristic of a population or process and statistic to describe a numerical characteristic of a sample x The symbol S to represent the probability of success for a process or the population proportion and p ˆ to represent a sample proportion of successes x The fundamental notion of sampling variability and how to simulate empirical sampling (null) distributions “by hand” (e.g., using cards) and using technology (e.g., with an applet) x How to estimate and interpret a p-value x
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