● Final exam Wednesday, 6/10, 11:30-2:30. ● Bring your own blue books ● Closed book. Calculators and 2-page cheat sheet allowed. No cell phone/computer. ● Exam stresses understanding; Format similar to midterm ● Cumulative, but emphasizes materials since the midterm Final Review Session

What We Have Done (Since Midterm) ● (Probability and probability distribution---this is before midterm, but discuss a bit anyway) ● The normal distribution ● Sampling distributions (many are normal) ● Inference about population parameters using sampling distributions: confidence interval construction and hypothesis testing ● Relationships: Scatter plots and correlation; Regression analysis; Two way tables

Probability and Probability Distributions ● Relative frequency probability and subjective probability ● Probability rules ● Probability distributions; Continuous and discrete distributions; Density curves ● Expectation of a probability distribution; Law of large numbers (sample means approaches population mean as N goes large)

● Knowing the mean and the standard deviation of a normal density curve, we should be able to compute the Z score, and should know how to find, if given a normal distribution table: – The probability the variable takes any range of values – At what percentile is a given value – What value corresponds to a given percentile ● In particular, some useful empirical rules: – 68% of the values fall within one standard deviation of the mean – 95% of the values fall within two standard deviations of the mean – 99.7% of the values fall within three standard deviations of the mean ● The normal distribution

Sampling Distributions; Confidence Intervals ● Distributions of sample statistics in repeated sampling. For sample mean, proportion, and difference in means, the sampling distributions are approximately normal. ● The characteristics of the sampling distribution (such as the mean) typically involve the corresponding population parameters. ● Using the properties of the normal distribution we can make statements about what range of values the sampling statistic can take with what probability. ● Such statements easily translate into statements about the population parameters, which we call confidence intervals.

Hypothesis Testing ● Logic: what's the probability of observing the sample data, if the hypothesis about the population parameter were true? If this probability is very small, we reject the hypothesis. Otherwise we fail to reject it. ● Elements: – Determine the two hypotheses test statistic – Compute the observed value of the test statistic, which is some sample statistic that has a known sampling distribution under the null hypothesis. – Using this sampling distribution, determine how (un)likely it is to observe the test statistic taking values as extreme as observed or more so, if the null hypothesis were true (i.e., find the “p-value”) – Make a decision/conclusion based on the p-value: do we consider the computed p-value “small” enough? – Significance level, Type I/Type II Error.

Scatter Plots and Correlation ● Graphical description: scatter plots – Look for direction, form, strength, and outliers ● Numerical measure: correlation coefficient – Definition – Direction and strength of linear relationships ● Significance, effect size, and explanatory power

Linear Regression ● The OLS estimation principle and properties of the OLS estimator ● Using estimated model for prediction and interpretation; Marginal effects ● Hypothesis testing: Is there a relationship? Is the model useful at all? (t-tests and F-tests) ● Goodness of fit, R-square ● Dummy variables and interaction effects ● Outliers; Extrapolation; Causation

Two Way Tables and the Chi-Square Test ● Is there a relationship between two categorical variables? ● Reasoning: If the null hypothesis were true, what should be the expected counts in the table cells? Are the differences between the observed and the expected counts due to random chance? Or are they “significant” so that we would reject the null hypothesis? ● The test statistic involves all the differences between the observed counts and the expected counts, and follows a Chi-Square distribution.

Q&A Thanks & Good Luck!