4. A new diagnostic test is developed for a rare but serious disease.The test is designed to quickly identify patients who are infected, allowing for rapid treatment intervention Medical researchers use the following hypotheses at alpha =0.01 to determine if a patient has the disease: H_(0) : The patient does not have the disease. H_(a) : The patient does have the disease. a. Describe a Type I and Type II error in context. b. Which would be more serious in this context, a Type I error or a Type II error? Explain. c. What is the probability the researchers will make a Type I error?

a. In this context, a Type I error would occur if the researchers incorrectly reject the null hypothesis ($H_0$) and conclude that a patient has the disease when they actually do not. This would result in a false positive diagnosis.<br /><br />A Type II error would occur if the researchers fail to reject the null hypothesis ($H_0$) and conclude that a patient does not have the disease when they actually do. This would result in a false negative diagnosis.<br /><br />b. In this context, a Type II error would be more serious than a Type I error. This is because a Type II error would mean that a patient with the rare but serious disease is not detected and therefore not treated promptly. This could lead to severe consequences for the patient's health and potentially even death.<br /><br />c. The probability of making a Type I error is determined by the significance level ($\alpha$) chosen by the researchers. In this case, the researchers have chosen $\alpha = 0.01$, which means they are willing to accept a 1% chance of incorrectly rejecting the null hypothesis and concluding that a patient has the disease when they do not.