1. Ch 11: Correlations (pt. 2) and Ch 12: Regression (pt.1) Nov. 13, 2014

2. Hypothesis Testing for Corr Same hypothesis testing process as before: 1) State research & null hypotheses – –Null hypothesis states there is no relationship between variables (correlation in pop = 0) –Notation for population corr is rho (  ) –Null:  = 0 (no relationship betw gender & ach) –Research hyp:  doesn’t = 0 (there is a signif relationship betw gender & ach)

3. (cont.) The appropriate statistic for testing the signif of a correlation (r) is a t statistic Formula changes slightly to calculate t for a correlation: Need to know r and sample size

4. Find the critical value to use for your comparison distribution – it will be a t value from your t table, with N-2 df Use same decision rule as with t-tests: –If (abs value of) t obtained > (abs value) t critical  reject Null hypothesis and conclude correlation is significantly different from 0.

5. Example For sample of 35 employees, correlation between job dissatisfaction & stress =.48 Is that significantly greater than 0? Research hyp: job dissat & stress are significantly positively correlated (  > 0) Null hyp: job dissat & stress are not correlated (  = 0) Note 1-tailed test, use alpha =.05

6. Regression Predictor and Criterion Variables Predictor variable (X) – variable used to predict something (the criterion) Criterion variable (Y) – variable being predicted (from the predictor!) –Use GRE scores (predictor) to predict your success in grad school (criterion)

7. Prediction Model Direct raw-score prediction model –Predicted raw score (on criterion variable) = regression constant plus the result of multiplying a raw-score regression coefficient by the raw score on the predictor variable –Formula b = regression coefficient (not standardized) a = regression constant

8. The regression constant ( a ) –Predicted raw score on criterion variable when raw score on predictor variable is 0 (where regression line crosses y axis) Raw-score regression coefficient ( b ) –How much the predicted criterion variable increases for every increase of 1 on the predictor variable (slope of the reg line)

9. Correlation Example: Info needed to compute Pearson’s r correlation xy(x-Mx)(x-Mx) 2 (y-My)(y-My) 2 (x-Mx)(y-My) 662.45.76244.8 12-2.66.76-245.2 561.41.96242.8 34-.6.36000 32-.6.36-241.2 Mx= 3.6 My= 4.0 0SSx= 15.2 0SSy= 16SP = 14.0 Refer to this total as SP (sum of products)

10. Formulas for a and b First, start by finding the regression coefficient (b): Next, find the regression constant or intercept, (a): This is known as the “Least Squares Solution” or ‘least squares regression’

11. Computing regression line (with raw scores) 6 1 2 5 6 3 4 3 2 X Y 14.015.2016.0 SS Y SS X SP mean 3.64.0 Ŷ =.688 +.92(x)

12. Interpreting ‘a’ and ‘b’ Let’s say that x=# hrs studied and y=test score (on 0-10 scale) Interpreting ‘a’: –when x=0 (study 0 hrs), expect a test score of.688 Interpreting ‘b’ –for each extra hour you study, expect an increase of.92 pts

13. Correlation in SPSS Analyze  Correlate  Bivariate –Choose as many variables as you’d like in your correlation matrix  OK –Will get matrix with 3 rows of output for each combination of variables Notice that the diagonal contains corr of variable with itself, we’re not interested in this… 1 st row reports the actual correlation 2 nd row reports the significance value (compare to alpha – if < alpha  reject the null and conclude the correlation differs significantly from 0) 3 rd row reports sample size used to calculate the correlation

14. Simple Regression in SPSS –Analyze  Regression  Linear –Note that terms used in SPSS are “Independent Variable” (this is x or predictor) and “Dependent Variable” (this is y or criterion) –Class handout of output – what to look for: “Model Summary” section - shows R 2 ANOVA section – 1 st line gives ‘sig value’, if <.05  signif –This tests the significance of the R 2 for the regression. If yes  it does predict y) Coefficients section – 1 st line gives ‘constant’ = a (listed under ‘B’ column) –Other line gives ‘unstandardized coefficient’ = b –Can write the regression/prediction equation from this info…