Chapter 7: Hypotheses Testing I. Sample and Populations A. Sample – subset of a the larger group

1. Chapter 7: Hypotheses Testing I. Sample and Populations A. Sample – subset of a the larger group B. Population – the larger group Sample Selection – select in such a way that is random, whereby each and every population member is equally likely to be selected and sample members roughly resemble the larger population ---over/under sampling produces sampling error, or the distance between estimated values of the population based on the sample and the true population values. • Null Hypothesis – a statement that predicts equal relationships or no relationship between two variables. Example: “There is no relationship between one’s likelihood of voting and one’s level of education.” And “There is no difference between the average score of 9th graders and 12th graders on basic math tests.” In other words, a null hypothesis says that two or more things are equal to, or unrelated to, each other. Observed differences are due to chance.

2. Research Hypothesis The Null states that there will be no relationship between variables, the research hypothesis states that there is a relationship and what the relationship is. It is a statement of expectation. Example: One’s education level is positively related to one’s likelihood of voting. Or “There is a difference between the average score of 9th graders and 12th graders on basic math tests.” Question for research = Which do we accept and which do reject? Key Differences: • Null refers to no relationship (equality) while the research hypo refers to an expected relationship (inequality). • Null always refers to population; research always refers to the sample. • Null hypo’s written in Greek letters while research hypo’s written in Roman letters. Null H0 : μ12 + μ9 Research Hypos: non-directional (p. 126); directional (p. 127) *One and Two-tailed tests

3. Components of a good hypothesis: • Not a question, stated in declarative form (no maybes). • Posits expected relationship (no fishing expedition). Sometimes can’t be done. • Reflect theory from which it emerges. It is linked to a larger body of literature. If theory is true, we should expect ____ relationship between X and Y. (theory example, free market theory and wealth; hypothesis example, positive relationship between state gdp and liberalized trade). If (theory), then (hypo); What you expect (hypo) and why you expect what you expect (theory) 4. Brief and to the point. 5. Testable (falsifiable). It is possible to accept or reject the null. Otherwise, meaningless.

4. Ch 6 Measurement Reliability and Validity Measurement – systematic observation and representation by numbers of the variables we are trying to examine. I. Measurement • Terms: 1. Operationalization – process by which we turn concepts (abstract idea) into indicators (concrete measurement scheme or numbers) 2. Concepts – abstract terms that we all employ to make sense of what we experience (state wealth). 3. Variables (indicators) – a concept that takes on a number of values (state per capita income). 4. Unit of analysis – the things (units) we are describing/analyzing (states). *A theory or thesis specifies how concepts relate; hypotheses specify how indicators or variables relate. B. Steps • Identify concepts (wealth, income, religiosity, etc.) • Select a measure or indicator of the presence, absence, or amount of these concepts using numbers. So, if you are examining the relationship between two concepts (literacy and democracy), then you need to find measures of those two concepts. • Examples of Political Measurement (religiosity, state policy-liberalism, individual conservatism, political participation, liberal judges, liberal courts, informed voter, SC adherence to stare decisis).

5. Confidence in Measurement – two threats to measurement confidence; they could be unreliable and/or invalid A. Reliability – the extent to which the measurement technique/indicator yields the same results given repeated trials (consistency) • Examples: a. Concept: my weight. Indicators: ask 20 people to guess it and a bathroom scale. Which is most reliable? b. Concept: political ideology over time. Indicators: are you a conservative, liberal, or moderate? Or ask them their opinion on 10 policy issues. Most reliable? • Validity – the extent to which the measurement technique/indicator actually measures (captures) the concept (accuracy). Examples: weight and guessing. Valid?

6. Chapter 8 (Probability) • Probability and Statistics A. Tells us likelihood of an outcome B. Tells us degree of confidence in a finding or outcome (i.e., how sure are we that the observed outcome is due to X versus random chance? AND how likely is it that our research hypothesis is true?). • Normal Curve or Bell-Shaped Curve Properties A. Mean, median and mode are same NOT Skewed

8. B. Perfectly symmetrical about the mean (i.e., two halves fit perfectly together). C. Tails of the normal curve are asymptotic. Curves come close, but never touch the horizontal axis. • Are curves usually normal? Yes, especially with large sets of data (more than 30). Most scores are concentrated in the center and few are concentrated at the ends (height, intelligence, p. 137). • Divisions of the Normal Curve (fig. 8.4) A. Mean is at the center B. Scores along x-axis correspond to standard deviations. C. Sections within the bell curve represent % of cases expected to fall therein. Geometrically true (these are percentages of entire normal distribution). D. For normal distributions (most data sets), practically all scores fall in between +3 and -3 sd’s (99.74%). Look at the probabilities of falling in between. -34.13% x 2 = 68.26% cases fall within 1 to -1 sd’s from mean.

10. Z-scores (standard scores; i.e., # of standard deviations from the mean) A. Allow us to compare distributions with one another because they are scores that are standardized in units of standard deviations (can’t compare scores if they are measured differently; nonsensical). Different variables or groups will have different means and cannot be compared. But z-scores between groups of data can be compared because they are equivalent (e.g., one unit above or below the mean, respectively).

11. Formula and interpretation (p. 142) *Do table on p. 143 • Comparing z-scores from different distributions -The raw scores of 12.8 and 64.8 in our data are equal distances from their respective means (z=.4 for both) • What z-scores represent A. Z-scores correspond to sections under the curve (percentages under the curve).

12. B. These percentages can be seen as probabilities of a certain score occurring given in Table B1 Appendix B. Example of what we are saying: “In a distribution with a mean of 100 and standard deviation of 10, what is the probability that any score will be 110 or above?” The answer = _________. C. What about a z-score of 1.38? What are the chances that a score will fall within the mean and a z-score of 1.38? _______ • What about above a z-score of 1.38?____ • What about at or below 1.38?______

13. What about between a z-score of 1 and 2.5? Answer:______ (look at picture p. 124) Again, we are asking, what is the probability that a score will fall in between 1 and 2.5 standard deviations (z’s) of the mean? -1 and 2.5? Computer help: probability computations on the web • Back to a research hypothesis Z-scores help us determine the likelihood of an event or outcome. Example: 1. Suppose we use a standard that if a coin lands on heads only 5% of the time, we can not have confidence in it (it is rigged). Not just chance.

14. 5% is the standard level for social science research. It means, that if the probability of an event (# of heads or differences between the averages of two groups, or so on) occurs in the extreme (i.e., defined as less than 5% of all cases), then it is unlikely (95% sure) that the outcome is due to random chance (options: it is random chance; it is due to something else: rigged, trickery). 2. Probability tells us how likely something is. It is highly unlikely that we would get 1 head out of ten flips of a coin. We might conclude that it is due to something else.

15. The null says that an outcome is likely (no differences between groups like heads or tails; any difference observed is due to random chance). But the research hypothesis says that the likelihood of an event is extreme and unlikely due to random chance alone. SO, if we find an extreme z-score (less than 5% chance of occurring), then we may say that observed differences are not due to chance but something else (independent variable). More to come! Null: outcomes equally as likely (diff due to chance) Research: Outcomes not equally as likely (diff due to something other than chance)