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 (round to three decimal places as needed)

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 76$ (number of correct results) and $n=98$. So, $\hat{p}=\frac{76}{98}\approx0.776$.

Step2: Calculate the test - statistic

The formula for the test - statistic $z$ in a one - sample proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$. Here, $p = 0.80$ (from the null hypothesis $H_0:p = 0.80$), $\hat{p}\approx0.776$, and $n = 98$.

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Step3: Calculate the P - value

Since this is a left - tailed test ($H_1:p\lt0.80$), the P - value is $P(Z\lt z)$. Using a standard normal table or a calculator with a normal distribution function, for $z=-0.60$, the P - value is $P(Z\lt - 0.60)=0.274$

Answer:

The P - value is $0.274$