## 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

- The assumptions are linearity, homoscedasticity, and normally distributed errors. See the text for more information.
- 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.
- t = (r) sqrt(N-2)/sqrt(1-r
^{2}) = (0.5) sqrt(18)/sqrt(1-.25) = 2.45 (This is significant at the .05 level.) - 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