Transcript Lecture 10
Variance Formula X 2 2 N X X 2 s 2 n 1 Probability A. The importance of probability Hypothesis testing and statistical significance Probabilistic causation - because error always exists in our sampling (sampling error) we can only deal with probabilities of being correct or incorrect in our conclusions The Normal Curve • The normal curve represents a probability distribution • 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 z-scores & conversions • What is a z-score? – A measure of an observation’s distance from the mean. – The distance is measured in standard deviation units. • • • • • If a z-score is zero, it’s on the mean. If a z-score is positive, it’s above the mean. If a z-score is negative, it’s below the mean. If a z-score is 1, it’s 1 SD above the mean. If a z-score is –2, it’s 2 SDs below the mean. Converting raw scores to z scores What is a z score? What does it represent Z = (x-µ) / σ Z = (710-563)/140 = 147/140 = 1.05 Converting z scores into raw scores X = z σ + µ [(1.05*140)+563=710] Examples of computing z-scores X X X X SD z X X SD 5 3 2 2 1 6 3 3 2 1.5 5 10 -5 4 -1.25 6 3 3 4 .75 4 8 -4 2 -2 The Normal Curve • A mathematical model or and an idealized conception of the form a distribution might have taken under certain circumstances. – A sample of means from any distribution has a normal distribution (Central Limit Theorem) – Many observations (height of adults, weight of children in Nevada, intelligence) have Normal distributions 9 Finding Probabilities under the Normal Curve So what % of GRE takers scored above and below 710? (Z = 1.05) - Why is this important? Infer the likelihood of a result Confidence Intervals/Margin of Error Inferential Statistics (to be cont.- ch.6-7) Stuff you don’t need to know: pi = ≈3.14159265 e = ≈2.71 Powers xa xb = x (a + b) xa ya = (xy)a (xa)b = x (ab) x(a/b) = bth root of (xa): Example X(1/2) = √X x(-a) = 1 / xa x(a - b) = xa / xb 13 THE NORMAL CURVE AND PROBABILITY Midterm • • • • Next Thursday Take home Babbie Chapters 1, 2, 3, 4, 5, 6, 13 Levin and Fox – Calculate variance and standard deviation – Calculate z-score