t Test for One Sample Why use a t test? The sampling distribution of t represents the distribution that would be obtained if a value of t were calculated for each sample mean for all possible random samples of a given size from some population. Just like the z sampling distribution!! Why use a t test? The figure indicates where the control and treatment group means are located. The question the t-test addresses is whether the means are statistically different. Why use a t test? When we are looking at the differences between scores for two groups, we have to judge the difference between their means relative to the spread or variability of their scores. Gas Mileage Investigation (269) Hypothesis test summary for t test for a population mean Research Problem Does the mean gas mileage for some population of cars drop below the legally required minimum of 45 mpg? Statistical Hypothesis H0: µ ≥ 45 H1: µ < 45 Decision Rule Reject H0 at the .01 level of significance if t≤ -3.365 (from Table B, Appendix C, given df = n-1 = 6 – 1 = 5). Calculations Given X = 43, sx = 0.89 (See table 13.1 on page 275 for computations) t = 43 – 45 = -2.25 0.89 Decision Retain H0 at .01 level of significance because t = -.2.25 is less negative than -3.365 Interpretation The population mean gas mileage could equal the required 45 mpg or more. The manufacturer shouldn’t be penalized. Table of critical values for t Table B of Appendix C page 520 Missing df If the desired number of degrees of freedom doesn’t appear in the df column of Table B, use the row in the table with the next smallest number of degrees of freedom. t ratio for a single population mean t = sample mean – hypothesized population mean estimated standard error or X - µhyp sx Estimating the standard error If the population standard deviation is unknown, it must be estimated from the sample. sx = _s_ √n Where s = SS √df Sum of squares for X SSx = Σ(X)2 – (ΣX)2 n Calculations for a t test 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Define n Sum all X scores Find mean of X Square each X score Sum the squared Xs Calculate SS Calculate s Calculate sx Assign value to µhyp Calculate t In class Progress Check 13.3 Confidence interval (p 276) X ± (tconf )(sx) Find value of tconf in Table B Homework Review question 13.7, a and b, due Tuesday 4/7 Show your work, answers in the back of the book.