Transcript Lecture 9
Hypothesis Testing • It is frequently expected that you have clear hypotheses when you have a study using quantitative data. • Older citizens are more likely to vote. Men are more likely to like computers. Rural schools perform higher than urban schools • How do we test these hypotheses? • It’s complicated. Regression, chi-square, ttest • But we need to start at the beginning The Normal Curve • The mean and standard deviation, in conjunction with the normal curve allow for more sophisticated description of the data and (as we see later) statistical analysis • For example, a school is not that interested in the raw GRE score, it is interested in how you score relative to others. • Even if the school knows the average (mean) GRE score, your raw score still doesn’t tell them much, since in a perfectly normal distribution, 50% of people will score higher than the mean. • This is where the standard deviation is so helpful. It helps interpret raw scores and understand the likelihood of a score. • So if I told you if I scored 710 on the quantitative section and the mean score is 591. Is that good? • It’s above average, but who cares. • What if I tell you the standard deviation is 148? • What does that mean? • What if I said the standard deviation is 5? • Calculating z-scores Converting raw scores to z scores What is a z score? What does it represent Z = (x-µ) / σ Converting z scores into raw scores X=zσ+µ Z = (710-563)/140 = 147/140 = 1.05 Finding Probabilities under the Normal Curve So what % of GRE takers scored above and below 710? Why is this important? Inferential Statistics (to be cont.) Why is the normal curve so important? • If we define probability in terms of the likelihood of occurrence, then the normal curve can be regarded as a probability distribution (the probability of occurrence decreases as we move away from the center – central tendency). • With this notion, we can find the probability of obtaining a raw score in a distribution, given a certain mean and SD (or standard error). Example of number 1: • President of UNLV states that the average salary of a new UNLV graduate is $60,000. We are skeptical and test this by taking a random sample of a 100 UNLV students. We find that the average is only $55,000. Do we declare the President a liar? Not Yet!!!! We need to make a probabilistic statement regarding the likelihood of the President’s statement. How do we do that? With the aid of the standard error of the mean we can calculate confidence intervals - the range of mean values within with our true population mean is likely to fall. How do we do that? • First, we need the sample mean • Second, we need the standard error, a.ka. standard deviation of the sampling distribution of means • Third, select a threshold to reject the null hypothesis: 5%, 10%, one-tail, two-tail (1.96 is usually the magic number for large samples) So is the President wrong? CI = Mean + or – 1.96 (SE) = 55,000 +/- 1.96 (3000) = 55,000 +/- 5880 = $49,120 to 60,880 We predict that the “true average salary” is within this range. Which means the President may be correct. Confidence Intervals for Proportions Calculate the standard error of the proportion: Sp = P1 P N 95% conf. Interval = P +/- (1.96)Sp Example • National sample of 531 Democrats (or Democratic-leaning) - Sept. 14-16, 2007 • Clinton 47%; Obama 25%; Edwards 11% • P(1-P) = .47(1-.47) = .47(.53) = .2491 • Divide by N = .2491/531 = .000469 • Square root of .000469 = .0217 • 95% CI = .47 +/- 1.96 (.0217) • .47 +/- .04116 or 0.429 to .511 • We are 95% confident the true population ranges from 42.9% to 51.1% Stone • Political system (politicians, public, special interests) not interested in testing hypotheses to get at causal relationships • Interested in causal stories • Actors are more likely to define the cause rather than “test hypotheses” or examine alternative explanations. • Causal stories often used to assign blame or deflect blame, gain support.