Statistics in Psychology: Estimation of Population Mean and Confidence Intervals, Slides of Statistics for Psychologists

Download Statistics in Psychology: Estimation of Population Mean and Confidence Intervals and more Slides Statistics for Psychologists in PDF only on Docsity! Introduction to Statistics in Psychology PSY 201 Lecture 22 Estimation of population mean Less than 5% of published psychological research is wrong. LAST TIME construct an interval around an observed statistic, X CI = statistic±(critical value) (standard error of the statistic) CI = X ± (tcv)(sX) 41 42 43 44 45 Sample Mean 0 0.2 0.4 0.6 0.8 2 CONFIDENCE we never say that a 95% confidence interval contains µ with probability 0.95 either the interval contains µ or it does not we can say that the procedure of producing CI’s produce intervals that contain µ with probability 0.95 we do talk about the confidence that an interval includes µ we would say that the confidence interval contains µ with confidence of 0.95 the confidence is in the procedure of calculating CIs 3 HYPOTHESIS TESTING remember SAT data: H0 : µ = 455 Ha : µ != 455 for n = 144, ! = 0.05, X = 535, tcv = ±1.9766, sX = 8.33, we reject H0 350 400 450 500 550 Sample Mean 0 0.01 0.02 0.03 0.04 4 CONFIDENCE INTERVAL given our data, we could also compute confidence intervals around X = 535 CI = X ± (tcv)(sX) CI95 = 535± (1.9766)(8.33) CI95 = (518.53, 551.46) 5 COMPARISON note: the rejected H0 : µ = 455 is consistent with the CI 455 is not in the 95% confidence interval (518.53, 551.46) CI contains only tenable values of µ, given the sampled data 6 docsity.com CI AND HYPOTHESIS TESTS CIs ask: which values of µ would it be reasonable for me to get the value of X that I found? Hypothesis tests ask: is the value of X I found consistent with a hypothesized value of µ? “reasonable” and “consistent” are defined relative to Type I error (!) 7 HYPOTHESIS TESTING constructing a CI is like testing a large number of non-directional hypotheses simultaneously: H0 : µ = 435 H0 : µ = 22 H0 : µ = 522 H0 : µ = 549 H0 : µ = 563 anything in the CI (518.53, 551.46) would be not be rejected, anything not in the CI would be rejected 8 EXAMPLE Take out a sheet of paper and write down the number of math-based courses you have taken at college (include physics, engineering, and computer science, if it was largely math-based) Now go around the room and sample this information from 10 other people Calculate the mean and standard deviation for your sample X = ! Xi n s = "#####$ ! i X2i " [( ! i Xi)2/n] n" 1 I’ll calculate the population mean for the class 9 HYPOTHESIS TEST (1) State the hypothesis: H0 : µ = 3 Ha : µ != 3 with ! = 0.05 (2) Set the criterion: with n = 10, df = 9, so t-distribution table gives tcv = 2.262 (3) Compute test statistic: Calculate standard error of the mean sX = s# n = s# 10 = t = X " 3 sX = (4) Make your decision: 10 CONFIDENCE INTERVAL CI = X ± (tcv)(sX) Calculate standard error of the mean sX = s# n = s# 10 = with n = 10, df = 9, so the t-distribution table gives tcv = 2.262 plug everything into your CI formula 11 COMPARISON Who rejected H0? Who have the value 3 outside their CI? Should be similar! What would change if you only sampled 4 scores instead of 10? 12 docsity.com