Assignment #4

 

Problem #1: (4) Sudden infant death syndrome (SIDS) causes babies to die suddently (often in their cribs) with no explanation. Deaths from SIDS have been greatly reduced by placing babies on their backs, but as yet, no cause is known. One sometimes-quoted rate for SIDS in a non-smoking, middle-class family is 1 in 8500. In a famous case in England in which two babies died of SIDS, the expert reported that the probability of this occuring was 1/72,250,000. How did the expert come up with this number? Because this is extremely unlikely, people with more than one case of SIDS have been convicted of murder. Many experts have since taken issue with this 1/72,250,000 calculation and argued that it is a flawed calculation. Why?

 

Problem #2: (4) A popular test for a certain medical condition has about a 3% rate of false positive results. That is, about 3% of people who do not have the disease will still have a positive test! If you give the test to 10 people, what is the likelihood of finding at least one false positive in the result? Hint: Review the in-class ‘0s in PIN numbers’ example.

 

Problem #3: (4) Internet sites often vanish or move so that references to them can’t be followed. In fact, one study claimed that 13% of Internet sites reference in major papers are lost within two years of publication. If a paper contains seven references, what is the probability that all seven references are still good two years later? What assumptions did you make in order to calculate this probability?

 

 Problem #4: (4) A study shows a strong positive correlation between the size of a hospital (measured by its number of beds), and the median number of days that patients remain in the hospital. Does this mean that you can shorten a hospital stay by choosing a smaller hospital? Explain.

 

Problem #5: (4) Census data shows tells us that about 13% of the population is a “senior” (i.e. 65 and older). We are also told that men make up about 49.2% of the population. What percentage of the population is female and senior?

 

Problem #6: (6) Medical Magnets:  Some claim that magnets can be used to reduce pain. Assume you are given a sample of 200 people to work with. Design the best experiment you can come up with to test this claim using the key principles we have discussed in class.

 

Problem #7: (8)

The 2000 census allowed each person to choose from a long list of races. That is, in the eyes of the Census Bureau, you belong to whatever race you say you belong to. If we choose a resident of the United States at random,. The 2000 census gives these probabilities:

 

Hispanic

Not-Hispanic

Asian

0.000

0.036

Black

0.003

0.121

White

0.060

0.691

Other

0.062

0.027

 Let A be the event that a randomly chosen American identifies himself/herself as Hispanic. Let B be the event that the person identifies as white.

a)      Verify that the table gives a legitimate assinment of probabilities.

b)      What is P(A)

c)       Describe Bc in words and find P(Bc)

d)      Express “the person chosen is a non-Hispanic white” in terms of events A and B. What is the probability of this event?

 

Problem #8: (6)

 

The Census Bureau reports that 27% of California residents are foreign-born. Suppose that you choose three Californians at random so that each has probability 0.27 of being foreign-born and the three are independent of each other.  Let the random variable W be the number of foreign-born people you choose.

a)      What are the possible values of W?

b)      Look at your three people in order. There are 8 possible arrangements of foreign (F) and domestic (D) birth. For example, FFD means the first two are foreign born and the third is not. What is the probability of each of the 8 arrangements?

c)       What is the value of W for each arrangement in B? What is the probability of each possible value of W? (This is the distribution of a Yes/No response for an SRS of size 3.