6. [24] Suppose that the time, T, from treatment to a recurrence for patients diagnosed with a disease is assumed to follow an Exponential distribution with a mean, 0 = 800 days. a. What is the probability that a patient will make it through 520 days without a recurrence? [2] b. If it has been 520 days since a patient has received treatment, without a recurrence, what is the probability that the patient will make it through the next 520 days without a recurrence? What property is being demonstrated here? [4] c. The data below give a data summary on the recurrence times for 80 patients who were treated for the disease. The average recurrence time for this sample of patients is f = 800. If the times are assumed to be independent from patient to patient, and the data are assumed to follow an exponential distribution with mean 0 = 800 days, give an expression for the probability of observing the data given in the table below. [3] Notes: A square bracket, or \"]\" implies that the value is included in the interval. A round bracket, or \"[\" implies that the value is not included in the interval. An expression is fine here. There is no need to compute an actual numerical answer. Time to Recurrence la 5 0, 4-00 4-00, 800 800, 1200 1200, 1600 > 1600 Number of Patients mm d. For this part, you will need to ll in the table given below. Determine the probability associated with each cell and compute the expected frequencies for each of the cells in the table. Compare the expected and observed values and answer the following question: Do you feel that the Exponential distribution with mean 800 provides an adequate description for these data? More formally, we are asking how well do you think the model ts the data? [10] *" Please determine the probability in the last cell (> 1600) to 4 d.p. by subtraction. This is often done in order to avoid any rounding issues. Time to Recurrence [Days] [0, 4-00] [400, 800] {800, 1200] (1200, 1600] > 1600 Observed Number of Patients Probability (to 4 d.p.] Expected Number of Patients (to 3 d.p.] Notes: Please use the probability value that you have given to 4 d.p. to calculate the expected number of patients, and give that result to 3 d.p. Based on the values in your table, determine if the model is a good fit for the data. We