Typhoid fever is more common in developing countries with poor sanitation and greater incidence of food contamination. The mean duration of this illness is approximately 4 weeks.14 Periodically, a team of doctors randomly selects 10 patients with typhoid fever and carefully measures the duration of the illness. If the mean duration is greater than 4.5 weeks, then a health alert is issued. Suppose the duration of this illness is normally distributed with standard deviation one week.
a. Find the probability that the sample mean duration is less than 4.5 weeks.
b. Find the probability that a health alert is issued.
c. Suppose the true mean duration has increased to 4.3 weeks. What is the probability that a health alert will be issued?
There are many regulations for catching lobsters off the coast of New England, including required permits, allowable gear, and size prohibitions. The Massachusetts Division of Marine Fisheries requires a minimum carapace length measured from a rear eye socket to the center line of the body shell. Any lobster measuring less than 31 4 inches must be returned to the ocean. For all lobster caught, suppose the carapace length is normally distributed with mean 41 8 inches and standard deviation 1 inch. A random sample of 15 lobsters is obtained.
a. Find the distribution of the sample mean carapace length.
b. What is the probability that the sample mean carapace length is more than 41 2 inches?
c. What is the probability that the sample mean carapace length is between 37 8 and 41 8 inches?
d. If the sample mean carapace length is less than 31 2 inches, a lobsterman will look for other places to set his traps. What is the probability that a lobsterman will be looking for a different location?