Several sections of an introductory Algebra class are being taught with the aid of computerized instruction. Students are given a quiz at the end of each chapter that they are required to pass. If they fail the first attempt, then they must attempt a computer-generated practice quiz online again and again until they pass it. Once they have passed the practice quiz, they are then allowed to retake the real quiz (a similar version to the first one). A random sample of 24 students that failed a quiz, but eventually retook it, was obtained. Let scores on a first failed attempt be population 1 , and the scores on a subsequent retake after passing a practice quiz be population 2. We want to see if the data shows that students, on average, will score higher on their second attempt after having passed the practice quiz. The sample produced the following statistics. \[ n=24, \bar{d} \approx-11.12, s_{d} \approx 14.322 \] Calculate the P-Value for the test. Round your answer using 4 decimal places.

Step 1 ：We are given a sample of 24 students who failed a quiz, but eventually retook it. The scores on a first failed attempt are considered as population 1, and the scores on a subsequent retake after passing a practice quiz are considered as population 2. We want to determine if the data shows that students, on average, will score higher on their second attempt after having passed the practice quiz. The sample produced the following statistics: \(n=24\), \(\bar{d} \approx-11.12\), \(s_{d} \approx 14.322\).

Step 2 ：We first calculate the t-statistic using the formula \(t = \frac{\bar{d}}{s_{d} / \sqrt{n}}\). Substituting the given values, we get \(t = \frac{-11.12}{14.322 / \sqrt{24}}\), which gives us a t-statistic of approximately -3.8037042228388405.

Step 3 ：We then calculate the degrees of freedom using the formula \(df = n - 1\). Substituting the given value of n, we get \(df = 24 - 1 = 23\).

Step 4 ：Next, we calculate the p-value. Since this is a two-tailed test, we multiply the result of the survival function by 2. Using the calculated t-statistic and degrees of freedom, we get a p-value of approximately 0.0009.

Step 5 ：Finally, we round the p-value to 4 decimal places to get \(\boxed{0.0009}\). This is the P-Value for the test.