Inferential Statistics for b and r

This section starts with assumptions on the errors that are necessary for statistical inference. Then, it gives an example of a significance test for the slope. Finally, it talks about constructing confidence intervals for the slope and closes with a significance test for the correlation.

Answers


  1. The assumptions are linearity, homoscedasticity, and normally distributed errors. See the text for more information.

  2. Use the table in this section or the inverse t distribution calculator to find that the critical value is t(N-2) = t(10) = 2.23. The upper limit of the 95% CI = b + (t)(sb) = .8 + 2.23(.3) = 1.47.

  3. t = (r) sqrt(N-2)/sqrt(1-r2) = (0.5) sqrt(18)/sqrt(1-.25) = 2.45 (This is significant at the .05 level.)

  4. First, convert r to z' (so .75 -> .973). The standard error of z' is 1/sqrt(N-3) = .213. Lower limit of CI = .973 - 1.96(.213) = 0.556. Now convert back from z' to r. r = .505