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 005 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.
$\begin{array}{l} H_{0} - p = 0.20 \\ H_{1} . p \neq 0.20 \end{array}$
C.
$\begin{array}{l} H_{0} : p = 0.80 \\ H_{1} : p < 0.80 \end{array}$

E
$\begin{array}{l} H_{0} : p = 0.80 \\ H_{1} : p \neq 0.80 \end{array}$
B.
$\begin{array}{l} H_{0} p = 0.80 \\ H_{1} p > 0.80 \end{array}$
D.
$\begin{array}{l} H_{0} p = 0.20 \\ H_{1} p > 0.20 \end{array}$

Fit.
$\begin{array}{l} H_{0}, p = 0.20 \\ H_{1} . p < 020 \end{array}$

The test statistic is $z = ◻$ (Rourh to two decimal places as needed)

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Answer

So, the test statistic is $z = - 0.74$ .

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Step 1 ：The null hypothesis (H0) is that the proportion of correct polygraph results is 80% or more, and the alternative hypothesis (H1) is that the proportion of correct polygraph results is less than 80%. So, the hypotheses are: $H_{0} : p = 0.80$ $H_{1} : p < 0.80$

Step 2 ：Next, we calculate the test statistic, which is a z-score (z). The formula for the z-score is: $z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} * (1 - p_{0})}{n}}}$ where $\hat{p}$ is the sample proportion, $p_{0}$ is the proportion in the null hypothesis, and n is the sample size.

Step 3 ：In this case, $\hat{p} = \frac{73}{97} = 0.7526$ , $p_{0} = 0.80$ , and n = 97.

Step 4 ：Substituting these values into the z-score formula, we get: $z = \frac{0.7526 - 0.80}{\sqrt{\frac{0.80 * (1 - 0.80)}{97}}} = \frac{- 0.0474}{\sqrt{0.0041}} = \frac{- 0.0474}{0.0640} = - 0.74$ (rounded to two decimal places).

Step 5 ：So, the test statistic is $z = - 0.74$ .

Answer

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