# Chapter 5. Research Questions About One Group

- Three factors help determine whether an observed estimate, such as the mean, is different from a norm: the size of the difference, the degree of variability, and the sample size.
- The
*t*distribution is similar to the*z*distribution, especially as sample sizes exceed 30, and*t*is generally used in medicine when asking questions about means. - Confidence intervals are common in the literature; they are used to determine the confidence with which we can assume future estimates (such as the mean) will vary in future studies.
- The logic behind statistical hypothesis tests is somewhat backwards, generally assuming there is no difference and hoping to show that a difference exists.
- Several assumptions are required to use the
*t*distribution for confidence intervals or hypothesis tests. - Tests of hypothesis are another way to approach statistical inference; a somewhat rigid approach with six steps is recommended.
- Confidence intervals and statistical tests lead to the same conclusions, but confidence intervals actually provide more information and are being increasingly recommended as the best way to present results.
- In hypothesis testing, we err if we conclude there is a difference when none exists (type I, or α, error), as well as when we conclude there is not difference when one does exists (type II, or β, error).
- Power is the complement of a type II, or β, error: it is concluding there is a difference when one does exist. Power depends on several factors, including the sample size. It is truly a key concept in statistics because it is critical that researchers have a large enough sample to detect a difference if one exists.
- The
*P*value first assumes that the null hypothesis is true and then indicates the probability of obtaining a result as or more extreme than the one observed. In more straightforward language, the*P*value is the probability that the observed result occurred by chance alone. - The
*z*distribution, sometimes called the*z*approximation to the binomial, is used to form confidence intervals and test hypotheses about a proportion. - The width of confidence intervals (CI) depends on the confidence value. 99% CI are wider than 95% CI because 99% CI provide greater confidence.
- Paired, or before-and-after, studies are very useful for detecting changes that might otherwise be obscured by variation within subjects, because each subject is his or her own control.
- Paired studies are analyzed by evaluating the differences
themselves. For numerical variables, the paired
*t*test is appropriate. - The kappa κ statistic is used to compare the agreement between two independent judges or methods when observations are being categorized.
- The McNemar test is the counterpart to the paired
*t*test when observations are nominal instead of numerical. - The sign test can be used to test medians (instead of means) if the distribution of observations is skewed.
- The Wilcoxon signed rank test is an excellent alternative
to the paired
*t*test if the observations are not normally distributed. - To estimate the needed sample size for ...