Okay, let's break this down step-by-step:
We have data on whether a subject lied (L) or told the truth (T), and whether the polygraph test indicated they lied (L) or told the truth (T).
So we have 4 possible outcomes:
LL: Subject lied, test indicated lied
LT: Subject lied, test indicated truth
TL: Subject told truth, test indicated lied
TT: Subject told truth, test indicated truth
We want to test the null hypothesis that a subject's truthfulness is independent of the polygraph test result.
So we need to calculate a test statistic that would allow us to determine if the observed frequencies of the 4 outcomes deviate significantly from what we would expect if the null hypothesis is true.
A good test for this is the chi-square test of independence. Here are the steps:
1) Calculate the expected frequency for each cell, assuming independence. This is (row total * column total) / total sample size.
2) Calculate the observed frequency for each cell from the data.
3) Square the difference between observed and expected for each cell.
4) Sum the squared differences across all cells. This gives you the chi-square statistic.
5) Compare the chi-square statistic to the critical value for 3 degrees of freedom at your desired alpha level (typically 0.05).
If the chi-square statistic exceeds the critical value, we reject the null hypothesis of independence. Otherwise, we fail to reject it.
Does this make sense? Let me know if you have any other questions! I can also walk you through an example if this would be helpful.