A patient’s blood cholesterol level is often checked during a routine physical examination. A blood sample is taken (from the finger or arm) and tested for total cholesterol and HDL (high-density lipoprotein, the “good cholesterol”) cholesterol levels in milligrams per deciliter (mg/dL). If total cholesterol is less than 200 mg/ dL, then no action is taken. If a patient’s total cholesterol is approximately 300 mg/dL, a new drug together with a strict diet is prescribed to reduce this number to a safe level. The amount of time (in days) it takes to reduce total cholesterol to a safe level has an exponential distribution with l 5 1/15. A patient with total cholesterol of 300 mg/dL is randomly selected and placed on this drug-and-diet regimen.
a. Find the mean number of days until the total cholesterol level is safe (less than 200 mg/dL).
b. Find the probability that the total cholesterol level will be safe in a number of days within two standard deviations of the mean.
c. Find a value d such that 75% of all such patients have a safe cholesterol level with d days.
d. Suppose two patients are selected at random. Find the probability that this drug-and-diet regimen works for at least one of the patients within 10 days.
A toy manufacturer routinely tests new toys in a controlled environment before deciding whether to actually market a toy. Research has shown that the amount of time (in minutes) a randomly selected child plays with a new toy has an exponential distribution with l 5 0.05. Suppose a new toy is presented for study.
a. What is the probability that a randomly selected child will play with the toy for at most 10 minutes
b. What is the probability that a randomly selected child will play with the toy for between 5 and 20 minutes?
c. Suppose a child plays with the toy for at least 15 minutes. What is the probability that the child will play with the toy for another 20 minutes?
d. If four different children each play independently with a new toy for at least 25 minutes, then the toy is immediately brought to market. What is the probability of this happening for a newly designed toy?