A 9-year-old girl did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under her hand without seeing it and without touching it. Among 271 trials, the touch therapists were correct 114 times. Use a 0.05 significance level to test the claim that touch therapists use a method equivalent to random guesses. Do the results suggest that touch therapists are effective? Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. \[ \begin{array}{l} H_{0}: p=0.5 \\ H_{1}: p<0.5 \end{array} \] C. \[ \begin{array}{l} H_{0}: p=0.5 \\ H_{1}: p>0.5 \end{array} \] B. \[ \begin{array}{l} H_{0}: p \neq 0.5 \\ H_{1}: p=0.5 \end{array} \] D. \[ \begin{array}{l} H_{0}: p=0.5 \\ H_{1}: p \neq 0.5 \end{array} \] Identify the test statistic for this hypothesis test. The test statistic for this hypothesis test is $\square$. (Round to two decimal places as needed.)

Step 1 ：Identify the null and alternative hypotheses for this test. The null hypothesis \(H_{0}\) is that the proportion of correct guesses by the touch therapists is equal to 0.5, which is what we would expect if they were guessing randomly. The alternative hypothesis \(H_{1}\) is that the proportion of correct guesses by the touch therapists is not equal to 0.5.

Step 2 ：Given values are: number of trials \(n = 271\), number of successes \(x = 114\), and proportion under the null hypothesis \(p_0 = 0.5\).

Step 3 ：Calculate the sample proportion \(p_{hat}\) by dividing the number of successes by the number of trials: \(p_{hat} = \frac{x}{n} = \frac{114}{271} \approx 0.421\).

Step 4 ：Calculate the test statistic \(z\) using the formula: \(z = \frac{p_{hat} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\). Substituting the given values into the formula gives \(z \approx -2.61\). This value is negative, indicating that the observed proportion of correct guesses by the touch therapists is less than what we would expect if they were guessing randomly.

Step 5 ：The final answer is: The null and alternative hypotheses for this test are: \(H_{0}: p=0.5\) and \(H_{1}: p \neq 0.5\). The test statistic for this hypothesis test is \(\boxed{-2.61}\).