Problem 1:
A medical testing laboratory saves money by combining blood samples for tests, so that only one test is conducted for several people. The combined sample tests positive if at least one person is infected. If the combined sample tests positive, then individual blood tests are performed. In a test for gonorrhea, blood samples from 40 randomly selected people are combined. Based on data from the Center for Disease Control, in a population at risk, the probability of a randomly selected person having gonorrhea is 0.03.
(a) Define the random variable in this study and its probability distribution
(b) What is the probability that a combined sample tests positive (i.e. at least one of the 40 randomly selected people have gonorrhea)?
Problem 2:
Suppose you live and study in Kathmandu, Africa and you will be driving home as soon as this class is over. After a few miles you feel tired of driving and you miss being in the biostatistics class. Therefore, you decide to stop and hang out on a bridge and collect data on the running speed of gazelles passing below. After four hours of data collection you have a normal curve of speeds with a mean of 55 mph and standard deviation of 7 mph. To delay your trip even more, you begin asking interesting questions as described below.
(a) Define the random variable in this study and its probability distribution.
(b) What is the probability that a gazelle runs at a speed greater than 70 mph?
(c) What is the running speed that corresponds to the slowest 20% of gazelles?
Problem 3:
Suppose patients arrive at the Penn Hospital ER at a rate of 10 per hour, on average.
(a) Define the random variable in this study and its probability distribution.
(b) What is the probability that 5 patients arrive in one hour?
(c) What is the probability that more than 12 patients arrive in one hour?
Problem 4:
In a length of hospitalization study conducted by several cooperating hospitals, a random sample of 24 peptic ulcer patients was drawn from a list of all peptic ulcer patients ever admitted to the participating hospitals and the length of hospitalization per admission was determined for each. The mean length of hospitalization per admission was found to be 8.15 with a standard deviation of 3 days. Assume that the length of hospitalization of peptic ulcer patients is approximately normally distributed.
(a) Construct and interpret a 90% confidence interval for the true length of stay for peptic ulcer patients.
(b) Based on other studies we can say that the true length of stay for duodenal ulcer patients is 9 days.
Do we have sufficient evidence in the data to conclude that the length of stay for peptic ulcer patients is significantly lower than duodenal ulcer patients? (Define the parameter(s), state the hypotheses, fix alpha, compute the test statistic and the p-value, and state the decision and conclusion in the context of the problem).