Answer :
Certainly! Let's follow the detailed, step-by-step solution for this hypothesis testing problem.
### Step 1: Identify the Hypotheses
Given the problem, we have to test whether the proportion of correct polygraph results is less than 80%. The null and alternative hypotheses can be defined as:
- Null hypothesis ([tex]\(H_0\)[/tex]): The population proportion of correct results is 80%.
[tex]\[ H_0: p = 0.80 \][/tex]
- Alternative hypothesis ([tex]\(H_1\)[/tex]): The population proportion of correct results is less than 80%.
[tex]\[ H_1: p < 0.80 \][/tex]
This set matches option [tex]\( \boxed{A} \)[/tex].
### Step 2: Collect the Sample Data
From the problem:
- Number of trials ([tex]\(n\)[/tex]) = 97
- Number of correct results = 75
- Number of wrong results = 22
### Step 3: Calculate the Sample Proportion
The sample proportion ([tex]\(\hat{p}\)[/tex]) of correct results is given by:
[tex]\[ \hat{p} = \frac{\text{Number of correct results}}{\text{Total number of trials}} = \frac{75}{97} \approx 0.7732 \][/tex]
### Step 4: Calculate the Standard Deviation of the Sampling Distribution
The standard deviation ([tex]\(\sigma_{\hat{p}}\)[/tex]) of the sampling distribution for the sample proportion is calculated using:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
where [tex]\(p_0\)[/tex] is the claimed population proportion (0.80), and [tex]\(n\)[/tex] is the sample size (97).
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.80 \cdot (1 - 0.80)}{97}} \approx 0.0406 \][/tex]
### Step 5: Calculate the Test Statistic (z)
The test statistic (z) for a sample proportion is given by:
[tex]\[ z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} \][/tex]
Plugging in the values we have:
[tex]\[ z = \frac{0.7732 - 0.80}{0.0406} \approx -0.66 \][/tex]
### Step 6: Calculate the P-value
The P-value corresponds to the probability that the sample proportion is less than the observed value under the null hypothesis. Using standard normal distribution tables or calculators:
[tex]\[ \text{P-value} \approx 0.255 \][/tex]
### Step 7: Decision Rule
We compare the P-value with the significance level ([tex]\(\alpha = 0.01\)[/tex]):
- If [tex]\( \text{P-value} < \alpha \)[/tex], we reject the null hypothesis.
- Otherwise, we do not reject the null hypothesis.
Since [tex]\(0.255\)[/tex] is greater than [tex]\(0.01\)[/tex], we do not reject the null hypothesis.
### Conclusion about the Null Hypothesis
We do not reject the null hypothesis ([tex]\(H_0: p = 0.80\)[/tex]).
### Final Conclusion About the Claim
Since we do not have sufficient evidence to reject the null hypothesis at the 0.01 significance level, we do not support the claim that the proportion of correct polygraph results is less than 80%. Therefore, based on the given data and at the 0.01 significance level, we do not have sufficient evidence to say that the polygraph results are correct less than 80% of the time.
Summary of Key Results:
- Sample proportion ([tex]\(\hat{p}\)[/tex]): [tex]\(0.7732\)[/tex]
- Standard deviation ([tex]\(\sigma_{\hat{p}}\)[/tex]): [tex]\(0.0406\)[/tex]
- Test statistic ([tex]\(z\)[/tex]): [tex]\(-0.66\)[/tex]
- P-value: [tex]\(0.255\)[/tex]
- Decision: Do not reject the null hypothesis
This is a thorough step-by-step solution to the problem.
### Step 1: Identify the Hypotheses
Given the problem, we have to test whether the proportion of correct polygraph results is less than 80%. The null and alternative hypotheses can be defined as:
- Null hypothesis ([tex]\(H_0\)[/tex]): The population proportion of correct results is 80%.
[tex]\[ H_0: p = 0.80 \][/tex]
- Alternative hypothesis ([tex]\(H_1\)[/tex]): The population proportion of correct results is less than 80%.
[tex]\[ H_1: p < 0.80 \][/tex]
This set matches option [tex]\( \boxed{A} \)[/tex].
### Step 2: Collect the Sample Data
From the problem:
- Number of trials ([tex]\(n\)[/tex]) = 97
- Number of correct results = 75
- Number of wrong results = 22
### Step 3: Calculate the Sample Proportion
The sample proportion ([tex]\(\hat{p}\)[/tex]) of correct results is given by:
[tex]\[ \hat{p} = \frac{\text{Number of correct results}}{\text{Total number of trials}} = \frac{75}{97} \approx 0.7732 \][/tex]
### Step 4: Calculate the Standard Deviation of the Sampling Distribution
The standard deviation ([tex]\(\sigma_{\hat{p}}\)[/tex]) of the sampling distribution for the sample proportion is calculated using:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
where [tex]\(p_0\)[/tex] is the claimed population proportion (0.80), and [tex]\(n\)[/tex] is the sample size (97).
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.80 \cdot (1 - 0.80)}{97}} \approx 0.0406 \][/tex]
### Step 5: Calculate the Test Statistic (z)
The test statistic (z) for a sample proportion is given by:
[tex]\[ z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} \][/tex]
Plugging in the values we have:
[tex]\[ z = \frac{0.7732 - 0.80}{0.0406} \approx -0.66 \][/tex]
### Step 6: Calculate the P-value
The P-value corresponds to the probability that the sample proportion is less than the observed value under the null hypothesis. Using standard normal distribution tables or calculators:
[tex]\[ \text{P-value} \approx 0.255 \][/tex]
### Step 7: Decision Rule
We compare the P-value with the significance level ([tex]\(\alpha = 0.01\)[/tex]):
- If [tex]\( \text{P-value} < \alpha \)[/tex], we reject the null hypothesis.
- Otherwise, we do not reject the null hypothesis.
Since [tex]\(0.255\)[/tex] is greater than [tex]\(0.01\)[/tex], we do not reject the null hypothesis.
### Conclusion about the Null Hypothesis
We do not reject the null hypothesis ([tex]\(H_0: p = 0.80\)[/tex]).
### Final Conclusion About the Claim
Since we do not have sufficient evidence to reject the null hypothesis at the 0.01 significance level, we do not support the claim that the proportion of correct polygraph results is less than 80%. Therefore, based on the given data and at the 0.01 significance level, we do not have sufficient evidence to say that the polygraph results are correct less than 80% of the time.
Summary of Key Results:
- Sample proportion ([tex]\(\hat{p}\)[/tex]): [tex]\(0.7732\)[/tex]
- Standard deviation ([tex]\(\sigma_{\hat{p}}\)[/tex]): [tex]\(0.0406\)[/tex]
- Test statistic ([tex]\(z\)[/tex]): [tex]\(-0.66\)[/tex]
- P-value: [tex]\(0.255\)[/tex]
- Decision: Do not reject the null hypothesis
This is a thorough step-by-step solution to the problem.
