# Chapter Summary

#### Chapter Objectives

12.1: Define normal distribution and z scores.
12.2: Explain confidence intervals and confidence levels.
12.3: Understand difference of means hypothesis testing with population data: z tests.
12.4: Understand difference of means hypothesis testing with sample data: t tests.
12.5: Explain hypothesis testing for a proportion.
12.6: Understand statistics with two samples.

• This chapter demonstrates how to use statistics.
• The normal distribution is often referred to as a bell curve because it is shaped like a bell, with most of the observations in the thick middle portion under the curve, and fewer observations in the tails.
• A z-score refers to the number of standard deviations by which a score deviates from the mean score.
• Confidence intervals are probability estimates of the true parameter value in terms of its occurrence between constructed boundaries.
• A confidence interval is an estimate of a range in which the population mean would likely be found. They can be calculated by using two different equations.
• The critical value indicates how much confidence we want in the confidence interval by using a confidence level. Researchers are free to choose any confidence level.
• Statistical hypotheses are statements about the value of a population parameter or about the relationship between two or more variables.
• Null hypotheses have two important characteristics:
• They are succinct and precise assertions about population parameters.
• They are stated in such a manner that data plus statistical theory allow us to reject them with a known degree of confidence that we are not making a mistake.
• In addition to stating a null hypothesis, researchers state another hypothesis called the research or alternative hypothesis.
• Hypotheses are accepted or rejected on the basis of statistical likelihood. A type 1 error occurs when a true null hypothesis is mistakenly rejected. A type 2 error occurs when a false null hypothesis is mistakenly accepted.
• We indicate how sure we want to be that we are not committing a type 1 error with the selection of an alpha level.
• A sampling distribution is a mathematical function that indicates the probability of different values of the estimator occurring.
• The term level of statistical significance is used to refer to the probability of making a type I error. The most common level of statistical significance in political science is .05.
• The steps for hypothesis testing requires various steps.
• The decision to reject or not reject the null hypothesis depends on the comparison between the observed test statistic and the critical value. There are two tests: one-tailed and two-tailed.
• All difference of means tests test hypotheses using the same logic. The basic logic is that researches want to determine if the difference between two means is different enough for them to conclude that there is a substantive and statistically meaningful difference.
• The z-test is a difference of means test for a population. The t-test is a difference of means test for a sample.
• Substantive interpretation is just as, if not more, important than statistical interpretation. To reinforce interpretation, remember that there are conditions that must be met in order to reject a null hypothesis and accept a research hypothesis.
• The basic tools allow researchers to make interesting analyses of data as well as understand basic analysis in the political science literature.