1-Sample t-testT-test purposeT formulaT distribution(cont.)Note on t-tableAnother noteExampleSlide 91-sample t test in SPSS1-Sample t-testMon, Apr 5thT-test purposeZ test requires that you know  from popUse a t-test when you don’t know the population standard deviation.One sample t-test:–Compare a sample mean to a population with a known mean but an unknown variance–Use Sy (sample std dev) to estimate  (pop std dev)T formulaT obtained = (ybar - y)Sy/sqrt NProcedure:–Compute t obtained from sample data –Determine cutoff point (not a z, but now a critical t) based on –Reject the null hypothesis if your observed t value falls in critical region (|t observed| > |t critical|)T distributionCan’t use unit normal table to find critical value – must use t table to find critical t–Based on degrees of freedom (df): # scores free to vary in t obtained–Start w/sample size N, but lose 1 df due to having to estimate pop std dev–Df = N-1–Find t critical based on df and alpha level you choose(cont.)To use the t table, decide what alpha level to use & whether you have a 1- or 2-tailed test  gives columnThen find your row using df.For  = .05, 2 tailed, df=40, t critical = 2.021–Means there is only a 5% chance of finding a t>=2.021 if null hyp is true, so we should reject Ho if t obtained > 2.021Note on t-tableWhen > 30 df, critical values only given for 40, 60, 120 df, etc.If your df are in between these groups, use the closest dfIf your df are exactly in the middle, be more conservative  use the lower dfExample: You have 50 df, choose critical t value given for 40 df (not 60).–You’ll use a larger critical value, making a smaller critical region  harder to find signifAnother noteNote that only positive values given in t table, so…If 1-tailed test,–Use + t critical value for upper-tail test (1.813)–Use – t critical value for lower-tail test (you have to remember to switch the sign, - 1.813)If 2-tailed test,–Use + and – signs to get 2 t critical values, one for each tail (1.813 and –1.813)ExampleIs EMT response time under the new system (ybar =28 min) less than old system ( = 30 min)? Sy = 3.78 and N=10–Ha: new < old ( < 30)–Ho: no difference ( = 30)–Use .05 signif., 1-tailed test (see Ha)–T obtained = (28-30) / (3.78 / sqrt10) = (28-30) / 1.20 = - 1.67(cont.)Cutoff score for .05, 1-tail, 9 df = 1.833–Remember, we’re interested in lower tail (less response time), so critical t is –1.833T obtained is not in critical region (not > | -1.833 |), so fail to reject nullNo difference in response time now compared to old system1-sample t test in SPSSUse menus for:Analyze  Compare Means  One sample tGives pop-up menu…need 2 things:–select variable to be tested/compared to population mean –Notice “test value” window at bottom. Enter the population/comparison mean here (use  given to you)–Hit OK, get output and find sample mean, observed t, df, “sig value” (AKA p value)–Won’t get t critical, but SPSS does the comparison for you…(if sig value < , reject