Questions: Trials in an experiment with a polygraph include 97 results that include 24 cases of wrong results and 73 cases of correct results. Use a 0.05 significance level to test the claim that such polygraph results are correct less than 80% of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method Use the normal distribution as an approximation of the binomial distribution. Let p be the population proportion of correct polygraph results. Identify the null and alternative hypotheses. Choose the correct answer below. A. H0 p=020 B. H0: p=0.80 H1-p>0.20 H1: p ≠ 0.80 C. H0: p=0.20 D. H0 · p=0.80 H1: p<0.20 H1: p<0.80 E. H0 · p=0.80 F. H0 p=0.20 H1: p>0.80 H1: p ≠ 0.20 The test statistic is z= (Round to two decimal places as needed.)

Trials in an experiment with a polygraph include 97 results that include 24 cases of wrong results and 73 cases of correct results. Use a 0.05 significance level to test the claim that such polygraph results are correct less than 80% of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method Use the normal distribution as an approximation of the binomial distribution. Let p be the population proportion of correct polygraph results. Identify the null and alternative hypotheses. Choose the correct answer below. A. H0 p=020 B. H0: p=0.80 H1-p>0.20 H1: p ≠ 0.80 C. H0: p=0.20 D. H0 · p=0.80 H1: p<0.20 H1: p<0.80 E. H0 · p=0.80 F. H0 p=0.20 H1: p>0.80 H1: p ≠ 0.20 The test statistic is z= (Round to two decimal places as needed.)
Transcript text: Trials in an experiment with a polygraph include 97 results that include 24 cases of wrong results and 73 cases of correct results. Use a 0.05 significance level to test the claim that such polygraph results are correct less than $80 \%$ of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P -value method Use the normal distribution as an approximation of the binomial distribution. Let p be the population proportion of correct polygraph results. Identify the null and alternative hypotheses. Choose the correct answer below. A. $\mathrm{H}_{0} \mathrm{p}=020$ B. $H_{0}: p=080$ $H_{1}-p>0.20$ $\mathrm{H}_{1}: \mathrm{p} \neq 0.80$ C. $\mathrm{H}_{0}: \mathrm{p}=0.20$ D. $H_{0} \cdot p=0.80$ $\mathrm{H}_{1}: \mathrm{p}<0.20$ $H_{1}: p<080$ E. $\mathrm{H}_{0} \cdot \mathrm{p}=0.80$ F. $\mathrm{H}_{0} \mathrm{p}=0.20$ $H_{1}: p>080$ $H_{1}: p \neq 0.20$ The test statistic is $z=$ $\square$ (Round to two decimal places as needed.)
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Solution

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Solution Steps

Step 1: Define Hypotheses

We define the null and alternative hypotheses as follows:

  • Null hypothesis: \( H_0: p = 0.80 \)
  • Alternative hypothesis: \( H_1: p < 0.80 \)
Step 2: Calculate Sample Proportion

The sample proportion of correct polygraph results is calculated as: \[ \hat{p} = \frac{73}{97} \approx 0.7526 \]

Step 3: Calculate Test Statistic

The test statistic \( Z \) is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values: \[ Z = \frac{0.7526 - 0.80}{\sqrt{\frac{0.80(1 - 0.80)}{97}}} \approx -1.1676 \] Rounding to two decimal places, we have: \[ Z \approx -1.17 \]

Step 4: Calculate P-value

The P-value associated with the test statistic \( Z \) is found to be: \[ \text{P-value} \approx 0.1215 \]

Step 5: Conclusion about the Null Hypothesis

We compare the P-value to the significance level \( \alpha = 0.05 \): Since \( 0.1215 > 0.05 \), we fail to reject the null hypothesis.

Step 6: Final Conclusion

Based on the results, we conclude that there is not sufficient evidence to support the claim that the polygraph results are correct less than \( 80\% \) of the time.

Final Answer

\(\boxed{\text{There is not sufficient evidence to support the claim that the polygraph results are correct less than } 80\% \text{ of the time.}}\)

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