In a study of 823 randomly selected medical malpractice lawsuits, it was found that 510 of them were dropped or dismissed. Use a 0.05 significance level to test the claim that most medical malpractice lawsuits are dropped or dismissed.
What is the P-value?
P-value $=0.000$
(Round to three decimal places as needed.)
What is the conclusion about the null hypothesis?
A. Fail to reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
B. Reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
C. Reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.
D. Fail to reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.

Step 1 ：Define the null hypothesis and the alternative hypothesis. The null hypothesis is that most medical malpractice lawsuits are not dropped or dismissed (i.e., less than or equal to 50% are dropped or dismissed). The alternative hypothesis is that most medical malpractice lawsuits are dropped or dismissed (i.e., more than 50% are dropped or dismissed).

Step 2 ：Use a one-sample proportion z-test to test the null hypothesis. The test statistic is calculated as follows: \(z = \frac{{p_{\text{hat}} - p_0}}{{\sqrt{{(p_0 * (1 - p_0)) / n}}}}\), where \(p_{\text{hat}}\) is the sample proportion, \(p_0\) is the hypothesized population proportion (in this case, 0.5), and n is the sample size.

Step 3 ：Calculate the P-value using the standard normal distribution. If the P-value is less than the significance level (0.05), reject the null hypothesis.

Step 4 ：Given that n = 823, x = 510, \(p_{\text{hat}} = 0.6196840826245443\), \(p_0 = 0.5\), z = 6.866988317601134, and p_value = 3.2785996140205498e-12.

Step 5 ：The P-value is extremely small, much less than the significance level of 0.05. This means that the probability of observing a sample proportion as extreme as the one we observed, assuming the null hypothesis is true, is virtually zero. Therefore, we have strong evidence to reject the null hypothesis.

Step 6 ：Final Answer: The P-value is \(\boxed{0.000}\).

Step 7 ：The conclusion about the null hypothesis is: C. \(\boxed{\text{Reject the null hypothesis because the P-value is less than or equal to the significance level, } \alpha}\).