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Transcript

Comparing Two Groups Statistics 2126 So far.. • We have been able to compare a sample mean to a population mean – z test – t test • Often times though we have two groups to compare • Is Group 1 different from Group 2 Matched pairs or correlated t test • AKA dependent sample t test • When subjects are matched on a variable or are used as their own controls, a sort of before and after thing if you will • Be very careful with this • But it is way powerful and easy to do Back to our mythical IQ course.. Before After Difference 103 98 -5 100 107 7 111 119 8 97 100 3 133 134 1 106 111 5 87 85 -2 A couple of summary statistics d 27 d d n 27 d 7 d 3.86 sd 3.48 Now it is a simple t test d t sd / n 3.86 0 t 3.48 / 7 3.86 t 3.48 / 2.65 3.86 t 1.31 t 2.95 And now for the decision • • • • t(6) = 2.447 tobt = 2.95 Reject H0 Our IQ course works!! Two sample problems • While is is useful to know how to compare a sample mean to a population mean and check for significance it is not all that common • We rarely know μ – Sometimes we do • IQ • Differences • Theoretical values The much more common question is… • Does one group differ from another? • Let’s say we had two classes with different teaching methods • Is there an effect of teaching method? Some (made up) data Statistic Class 1 Class 2 Mean 77 71 Standard deviation 6.2 6.7 Number of students 49 52 Our hypotheses • • • • • • Are the two classes different? H 0 μ1 = μ2 H A μ1 ≠ μ2 Or we could restate them like this: H 0 μ1 - μ2 = 0 H A μ1 ≠ μ2 ≠ 0 Let’s go back to the original t formula Statistic H0 ↓ ↓ x t s/ n ← Error Figure it out ( x1 x2 ) ( 1 2 ) t error practicall y.... ( x1 x2 ) t error Now about that error… • We cannot just add the values of s for each group • They must be weighted 2 1 2 2 s s n1 n2 6.2 2 49 6.7 2 52 So the formula is t x1 x2 2 1 2 2 s s n1 n1 Degrees of freedom • With a one sample t test we lose one degree of freedom • Because we calculated one standard deviation • Here was have calculated 2 • So we lose 2 df • In our case we have 99 df Sub in the values t 77 71 2 2 6.2 6.7 49 52 6 t 1.65 t 4.67 rejectH0 conclusions • All t tests are based on the same formula • Keep the assumptions in mind – SRS – Homogeneity of variance – Independence of observations