Question

Trials in an experiment with a polygraph include 97 results that include 22 cases of wrong results and 75 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.

A. Identify the null and alternative hypotheses.

B. What is the test​ statistic?
​(Round to two decimal places as​ needed.)

C. What is the​ P-value?
​(Round to four decimal places as​ needed.)

D. What is the conclusion about the null​ hypothesis?
A. Reject the null hypothesis because the​ P-value is greater than the significance​ level, alpha.
B. Fail to reject the null hypothesis because the​ P-value is greater than the significance​ level, alpha.
C.Fail to reject the null hypothesis because the​ P-value is less than or equal to the significance​ level, alpha.
D. Reject the null hypothesis because the​ P-value is less than or equal to the significance​ level, alpha.

E. What is the final​ conclusion?
A.There is not sufficient evidence to support the claim that the new method is effective in increasing the likelihood that a baby will be a girl.
B.There is not sufficient evidence to warrant rejection of the claim that the new method is effective in increasing the likelihood that a baby will be a girl.
C.There is sufficient evidence to warrant rejection of the claim that the new method is effective in increasing the likelihood that a baby will be a girl.
D. There is sufficient evidence to support the claim that the new method is effective in increasing the likelihood that a baby will be a girl.

Answer 1

The statistical software output for this problem is:

Hence,

A) Hypotheses:

Ho: p = 0.80

Ha: p < 0.80

B) Test statistic = -0.66

C) p - Value = 0.2546

D) Conclusion: Fail to reject the null hypothesis because the​ P-value is greater than the significance​ level, alpha. Option B is correct.

E) Final conclusion: There is not sufficient evidence to support the claim that the new method is effective in increasing the likelihood that a baby will be a girl. Option A is correct.

Answer 2

A. Null and Alternative Hypotheses: Null Hypothesis (H0): The proportion of correct polygraph results is greater than or equal to 80% (p ≥ 0.80). Alternative Hypothesis (Ha): The proportion of correct polygraph results is less than 80% (p < 0.80).

B. Test Statistic: To test the claim, we will use the z-test for proportions. The formula for the z-test statistic for proportions is:$�=\frac{\widehat{�}-�}{\sqrt{\frac{�(1-�)}{�}}}$where:

• $\widehat{�}$ is the sample proportion of correct results.

• $�$ is the hypothesized proportion under the null hypothesis.

• $�$ is the sample size.

We are given that there are 97 trials, out of which 75 are correct. So, the sample proportion of correct results is:$\widehat{�}=\frac{75}{97}$

C. P-value: The P-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. To find the P-value, we will use the standard normal distribution table or a statistical software tool. Since the P-value method is requested, we will use the z-table to find the P-value corresponding to the calculated z-test statistic.

D. Conclusion about the Null Hypothesis: We compare the calculated P-value to the significance level (α) of 0.05. If the P-value is less than or equal to α, we reject the null hypothesis; otherwise, we fail to reject it.

E. Final Conclusion: Based on the conclusion about the null hypothesis, we will make the final conclusion related to the original claim.

Please note that we need the sample size to calculate the test statistic and P-value. Since the sample size is not provided, I'm unable to proceed with the calculations. If you have the sample size information, please provide it, and I'll be happy to assist you further with the calculations and final conclusion.

Answer 3

A. Null and Alternative Hypotheses: Let p be the proportion of correct polygraph results. The null hypothesis (H0) and alternative hypothesis (Ha) can be stated as follows:

Null Hypothesis (H0): p ≥ 0.80 (Polygraph results are correct 80% of the time or more). Alternative Hypothesis (Ha): p < 0.80 (Polygraph results are correct less than 80% of the time).

B. Test Statistic: Since the sample size is large and we are using the normal distribution as an approximation of the binomial distribution, we can use the z-test statistic for proportions.

The formula for the z-test statistic is: z = (p̂ - p0) / √(p0 * (1 - p0) / n)

Where: p̂ is the sample proportion of correct polygraph results (75/97 in this case). p0 is the hypothesized proportion under the null hypothesis (0.80 in this case). n is the sample size (97 in this case).

C. P-value: To calculate the P-value, we need to find the probability that a z-score is less than the computed z-test statistic.

D. Conclusion about the Null Hypothesis: We will compare the P-value with the significance level (alpha) of 0.05. If the P-value is less than or equal to alpha, we reject the null hypothesis (H0). Otherwise, if the P-value is greater than alpha, we fail to reject the null hypothesis (H0).

E. Final Conclusion: Based on the conclusion from the test, we will make a final statement addressing the original claim.

Let's proceed with the calculations:

First, calculate the sample proportion: p̂ = 75/97 ≈ 0.7732

Next, calculate the z-test statistic: z = (0.7732 - 0.80) / √(0.80 * (1 - 0.80) / 97) z ≈ (-0.0268) / √(0.80 * 0.20 / 97) z ≈ -0.0268 / 0.0452 z ≈ -0.5929

Now, find the P-value corresponding to the z-test statistic -0.5929 from the standard normal distribution table or using a statistical tool: P-value ≈ 0.2778

D. Conclusion about the Null Hypothesis: Since the P-value (0.2778) is greater than the significance level (alpha = 0.05), we fail to reject the null hypothesis (H0).

E. Final Conclusion: The final conclusion is: B. There is not sufficient evidence to warrant rejection of the claim that the new method is effective in increasing the likelihood that a baby will be a gi