1. Statistics: Unlocking the Power of Data Lock 5 Section 6.12 Test for a Difference in Means

2. Statistics: Unlocking the Power of Data Lock 5 Outline Test for a difference in means

3. Statistics: Unlocking the Power of Data Lock 5 T-Test for a Difference in Means If the population is approximately normal or if sample sizes are large (n 1 ≥ 30, n 2 ≥ 30), the p- value can be computed as the area in the tail(s) beyond t of a t-distribution with degrees of freedom equal to the smaller of n 1 – 1 and n 2 – 1 H0:μ1=μ2H0:μ1=μ2

4. Statistics: Unlocking the Power of Data Lock 5 The Pygmalion Effect Source: Rosenthal, R. and Jacobsen, L. (1968). “Pygmalion in the Classroom: Teacher Expectation and Pupils’ Intellectual Development.” Holt, Rinehart and Winston, Inc. Pygmalion Effect: the greater the expectation placed upon people, the better they perform Teachers were told that certain children (chosen randomly) were expected to be “growth spurters,” based on the Harvard Test of Inflected Acquisition (a test that didn’t actually exist). The response variable is change in IQ over the course of one year.

5. Statistics: Unlocking the Power of Data Lock 5 The Pygmalion Effect ns Control Students2558.4212.0 “Growth Spurters”6512.2213.3 Does this provide evidence that the Pygmalion Effect exists? (that merely expecting a child to do better actually causes the child to do better?) (a) Yes (b) No *s 1 and s 2 were not given, so I set them to give the correct p-value

6. Statistics: Unlocking the Power of Data Lock 5 Pygmalion Effect We have evidence that positive teacher expectations significantly increase IQ scores, on average, in elementary school children. 1. State hypotheses: 2. Check conditions: 3. Calculate statistic: 5. Calculate test statistic: 7. Interpret in context: t with 65 – 1 = 64 df, upper tail p-value = 0.02 6. Compute p-value: H 0 :  1 =  2 H a :  1 >  2  1 = average change in IQ for “growth spurters”  1 = average change in IQ for control n 1 = 65 ≥ 30, n 2 = 255 ≥ 30 4. Calculate standard error:

7. Statistics: Unlocking the Power of Data Lock 5 Pygmalion Effect From the paper: “The difference in gains could be ascribed to chance about 2 in 100 times”