5. The prevalence of prostate cancer in the U.S. in 2013 was 0.9%. Two methods are commonly used to screen for prostate cancer: PSA (a blood test), and digital rectal exam (DRE).

Researchers found that the PSA test had a sensitivity of 67%. In other words, two-thirds of all the cases who truly have prostate cancer were detected. One-third of the cases of prostate cancer go undiagnosed. On the other hand, when the PSA test indicated no disease, in almost all cases there was no disease. The specificity was 97%.

DRE has a low sensitivity of 50%, and the specificity of 94% is also lower than PSA.

Calculate the positive predictive value (PPV) and negative predictive value (NPV) of each method. Explain in plain English the value of PPV and NPV.

6. Based on a study of sizes of metastatic carcinoid tumors in the heart (U. Panda et. al, 2002), assume that lengths of metastatic carcinoid tumors in the heart are normally distributed with a mean of 1.8 cm and standard deviation of 0.5 cm.

(a) Draw the normal distribution.

(b) What percent of metastatic carcinoid tumors do you expect to exceed 3 cm in length? Less than 3 cm?

(c) What percent of metastatic carcinoid tumors do you expect to be between 1 cm and 2 cm long? (d) What tumor length represents the 25th percentile? the 75th percentile?

(e) Estimate the IQR of tumor length.

(f) At least how long are the top 20% metastatic carcinoid tumor lengths?

(g) Below what value are the lowest 20% of metastatic carcinoid tumor lengths?

5.88 Motel Reservations. M. F. Driscoll and N. A. Weiss dis- cussed the modeling and solution of problems concerning mo- tel reservation networks in \"An Application of Queuing Theory to Reservation Networks\" (TIMS, Vol. 22, No. 5, pp. 540546). They dened a Type 1 call to be a call from a motel's computer terminal to the national reservation center. For a certain motel, the number, X, of Type 1 calls per hour has a Poisson distribu- tion with parameter = 1.7. Determine the probability that the number of Type 1 calls made from this motel during a period of 1 hour will be a. exactly one. b. at most two. (3. at least two. (Hint: Use the complementation rule.) (1. Find and interpret the mean of the random variable X. e. Determine the standard deviation of X