Political Science Research Study Guide Unit 3 Test Ch 6 7 Inference and Hypothesis Testing Chapter 6 Populations and Samples There are two types of data sets political scientists might use o Population data for every possible relevant case such as those from the U S census o Sample a subset of cases that is drawn from an underlying population There are random samples samples where every member of the population have an equal chance of being selected And then there are samples of convenience that are non random Statistical Inference the process of using what we know about the sample to make probabilistic statements about the broader population Basics of Probability Theory Outcome the result of a random observation Independent Outcomes two or more outcomes such that the realization of one of the outcomes does not affect the realization of another outcome We use probability theory to decide whether the patterns of relationships we observe in a sample could have occurred simply by chance Learning about the Population from a Sample The Central Limit Theorem Central Limit Theorem a fundamental result from statistics indicating that if one were to collect an infinite number of random samples and plot the resulting sample means those sample means would be distributed normally around the true population mean Applies only to samples that are selected randoml The Normal Distribution Normal Distribution a bell shaped statistical distribution that can be entirely characterized by its mean and standard deviation Properties of normal distribution o It is symmetrical about its mean meaning the mean median and mode are the same o The normal distribution has a predictable area under the curve within specified distances of the mean The 68 95 99 rule is a useful characteristic of the normal distribution which states that moving 1 2 and 3 standard deviations from the mean will leave 68 95 and 99 of the distributions area under the curve o It does not apply to any other shaped distribution Frequency Distribution a distribution of actual scores in the sample Sampling Distribution a hypothetical distribution of sample means Standard error of the mean the standard deviation of the sampling distribution of sample means o Its simply equal to the sample standard deviation divided by the square root of the sample size Confidence Interval a probabilistic statement about the likely value A Note on The Effects of Sample Size o As the formula for confidence interval indicates the smaller the standard errors the tighter our resulting confidence intervals will be o Larger standard errors will produce wider confidence intervals o Larger sample sizes will reduce the size of the standard errors o Smaller sample sizes will increase the size of the standard errors Chapter 7 Bivariate Hypothesis Testing o Bivariate hypothesis tests help us answer the question Are X and Y related Choosing The Right Bivariate Hypothesis Test o Tabular analysis a type of bivariate analysis that is appropriate when there are two categorical variables o Difference of Means Test a method of bivariate hypothesis testing that is appropriate when there is a categorical independent variable and a continuous dependent variable o Correlation Coefficient when both the dependent and independent variable are continuous o All Roads Lead to P o The P value stands for probability and ranges between 0 and 1 o It is the closest thing we have to a bottom line in statistics o P value the probability that we would see the relationship that we are finding because of random chance o The lower the p value the greater the confidence we have that there is a systematic relationship between the two variables for which we estimated the particular P value The Limitations of the P value o The logic of a P value is not reversible in other words P 001 does not mean that there is a 999 chance that something systematic is going on o It is also important to realize that even though our P value tells us something about our confidence that there is no relationship between two variable it does NOT tell us whether that relationship is causal o They do not directly reflect the quality of the measurement procedure for our o P values are always based on the assumption that you are drawing a perfectly random sample from the underlying population variables o Pi P i o The formula for this is have in our P value o The further we are from a truly random sample the less confidence we should From P values to Statistical Significance o Statistically Significant a conclusion based on observed data that the relationship between two variables is not due to random chance and therefore exists in the broader population o An assertion of statistical significance depends on a number of other factors o Statistical significance is achieved only to the extent that the assumptions underlying the calculation of the p value hold o There are a variety of different standards for what is a statistically significant p value o Most social scientists use the standard of a p value of 05 There are 3 Bivariate Hypothesis Tests 1 Tabular analysis 2 Difference of means 3 Correlation coefficient 1 Tabular analysis reading a table Any time that you see a table it is very important to take some time to make sure that you understand what is being conveyed in the table We can break this into the following three step process What are the variables that define the rows and columns of the table What do the individual cell values represent What are if any the general patterns you see in the table The chi squared 2 test for tabular association a statistical test for a relationship between two categorical variables The summation sign in this formula signifies that we sum over each cell in the table If all observed values were exactly equal to the values that we expect if there were no relationship between the two variables then X 2 0 Critical Value a predetermined standard for a statistical test such that if the calculated value is greater than the critical value then we conclude that there is a relationship between two variables and if the calculated value is less than the critical value we cannot make such a conclusion Degrees of Freedom the number of pieces of information we have beyond the minimum that we would need to make a particular inference 2 Difference of Means o In this type of bivariate hypothesis test we check if the means are different across the values of the independent variable o We follow the basic
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