To test the claim that polygraph results are correct less than 80% of the time, we'll perform a hypothesis test for the proportion of correct results. Here’s how you can do it step-by-step:

1. Identify the hypotheses:

   • Null Hypothesis ($H_0$): $p = 0.80$

     • This states that the proportion of correct results is 80%.

   • Alternative Hypothesis ($H_1$): $p < 0.80$

     • This states that the proportion of correct results is less than 80%.

2. Determine the sample proportion ($\hat{p}$):

   • Total trials: 97
   • Correct results: 73

   $\hat{p} = \frac{73}{97} \approx 0.7526$

3. Find the test statistic:

   • Use the formula for the z-test for proportions:

   $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

   Here, $p_0 = 0.80$ and $n = 97$.

   $z = \frac{0.7526 - 0.80}{\sqrt{\frac{0.80 \times (1 - 0.80)}{97}}}$

   $z = \frac{-0.0474}{\sqrt{\frac{0.80 \times 0.20}{97}}}$

   $z = \frac{-0.0474}{\sqrt{\frac{0.16}{97}}}$

   $z = \frac{-0.0474}{\sqrt{0.001649}}$

   $z = \frac{-0.0474}{0.0406} \approx -1.167$

4. Find the P-value:

   • Look up the z-value in a standard normal distribution table (or use a calculator) to find the P-value. The P-value for $z = -1.167$ (looking for the left tail because $H_1: p < 0.80$) is approximately 0.121.

5. Conclusion about the null hypothesis:

   • Compare the P-value with the significance level ($\alpha = 0.05$).

   • Since $0.121 > 0.05$, we fail to reject the null hypothesis.

6. Final conclusion:

   • There is not enough statistical evidence to support the claim that the polygraph results are correct less than 80% of the time at the 0.05 significance level.

In summary, based on this test, we cannot conclude that the polygraph results are less than 80% accurate.