LAST TIME CONFIDENCE construct an interval around an observed statistic X we never say that a 95 confidence interval contains with probability 0 95 Introduction to Statistics in Psychology PSY 201 either the interval contains or it does not Professor Greg Francis CI statistic critical value standard error of the statistic Lecture 22 CI X tcv sX Estimation of population mean Less than 5 of published psychological research is wrong we can say that the procedure of producing CI s produce intervals that contain with probability 0 95 we do talk about the confidence that an interval includes 0 8 we would say that the confidence interval contains with confidence of 0 95 0 6 the confidence is in the procedure of calculating CIs 0 4 0 2 0 41 42 43 44 45 Sample Mean 2 3 HYPOTHESIS TESTING CONFIDENCE INTERVAL COMPARISON remember SAT data given our data we could also compute confidence intervals around X 535 note the rejected H0 455 is consistent with the CI Ha 455 CI X tcv sX 455 is not in the 95 confidence interval 518 53 551 46 for n 144 0 05 X 535 tcv 1 9766 sX 8 33 we reject H0 CI95 535 1 9766 8 33 CI contains only tenable values of given the sampled data H0 455 CI95 518 53 551 46 0 04 0 03 0 02 0 01 0 350 400 450 500 550 Sample Mean 4 5 6 CI AND HYPOTHESIS TESTS CIs ask which values of 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 reasonable and consistent are defined relative to Type I error HYPOTHESIS TESTING EXAMPLE constructing a CI is like testing a large number of non directional hypotheses simultaneously 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 H0 435 H0 22 H0 522 H0 549 H0 563 anything in the CI 518 53 551 46 would be not be rejected anything not in the CI would be rejected Now go around the room and sample this information from 10 other people Calculate the mean and standard deviation for your sample X s i Xi n Xi2 i Xi 2 n n 1 I ll calculate the population mean for the class 7 8 9 HYPOTHESIS TEST CONFIDENCE INTERVAL COMPARISON CI X tcv sX Who rejected H0 1 State the hypothesis H0 3 Ha 3 with 0 05 Calculate standard error of the mean s s sX n 10 2 Set the criterion with n 10 df 9 so t distribution table gives tcv 2 262 with n 10 df 9 so the t distribution table gives tcv 2 262 3 Compute test statistic plug everything into your CI formula Who have the value 3 outside their CI Should be similar What would change if you only sampled 4 scores instead of 10 Calculate standard error of the mean s s sX n 10 t X 3 sX 4 Make your decision 10 11 12 STATISTICAL PRECISION STATISTICAL PRECISION STATISTICAL PRECISION consider the equation for confidence intervals since we can also change tcv by changing the level of confidence CI X tcv sX where X is the sample mean tcv is the critical value using the appropriate t distribution for the desired level of confidence sX is the estimated standard error of the mean s sX n s sX n increasing the sample size n produces narrow widths of CI narrower widths imply greater precision about where is located increasing n also modifies tcv by changing degrees of freedom df n 1 larger level of confidence implies smaller which implies larger tcv which implies larger width of CI makes sense we become more confident the interval includes by broadening the interval of course then we are less sure about the value larger df leads to smaller tcv see t distribution table smaller tcv or sX produce narrower widths 13 14 15 PUBLISHED DATA CONCLUSIONS NEXT TIME most researchers in the behavioral sciences use 0 05 estimation more hypothesis testing confidence intervals tests for correlation relationship with hypothesis testing Is there a correlation between homework and exam grades this means that they make a Type I error only 5 of the time or less no way to completely avoid making mistakes statistical precision this makes it quite likely that some of the data in published journals is wrong it is important in science to double and triple check everything if a bit of data is tremendously important better replicate the experimental finding 16 17 18