Question

trials in an experiment with a polygraph include 98 results that include 22 cases of wrong results and 76 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. ( h_0:p = 0.80 )
( h_1:p>0.80 )
b. ( h_0:p = 0.20 )
( h_1:p<0.20 )
c. ( h_0:p = 0.80 )
( h_1:p<0.80 )
d. ( h_0:p = 0.80 )
( h_1:p
eq0.80 )
e. ( h_0:p = 0.20 )
( h_1:p>0.20 )
f. ( h_0:p = 0.20 )
( h_1:p
eq0.20 )
the test statistic is ( z=-0.60 ) (round to two decimal places as needed)
the p - value is 0.274 (round to three decimal places as needed)

Explanation:

Step1: State the hypotheses

The claim is that polygraph results are correct less than 80% of the time. So, the null hypothesis \(H_0:p = 0.80\) (assuming no difference from the 80% claim initially) and the alternative hypothesis \(H_1:p<0.80\) (since we are testing for less - than).

Step2: Calculate the sample proportion \(\hat{p}\)

The sample size \(n = 98\), and the number of correct results \(x = 76\). Then \(\hat{p}=\frac{x}{n}=\frac{76}{98}\approx0.776\)

Step3: Calculate the test - statistic \(z\)

The formula for the test - statistic in a proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.80\), \(\hat{p}=0.776\), and \(n = 98\)

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Step4: Calculate the P - value

For a one - tailed (left - tailed) test with \(z=-0.60\), using the standard normal distribution table or a calculator, the P - value is \(P(Z < - 0.60)\)
Looking up in the standard normal table: \(P(Z < - 0.60)=0.274\)

Step5: Make a conclusion about the null hypothesis

Since the significance level \(\alpha = 0.05\) and \(P - value=0.274>0.05\), we fail to reject the null hypothesis.

Step6: Make a final conclusion

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

Answer:

• Null hypothesis: \(H_0:p = 0.80\)
• Alternative hypothesis: \(H_1:p<0.80\)
• Test statistic: \(z=-0.60\)
• P - value: \(0.274\)
• Conclusion about the null hypothesis: Fail to reject \(H_0\)
• Final conclusion: There is not sufficient evidence to support the claim that polygraph results are correct less than 80% of the time.