Mean and Standard Deviation for Medical Issues

Because they are so easy to correct, multiple choice questions are commonly used for class test, SAT test, MCAT test for medical schools and many other circumstances. The table listed describes the probability distribution for the number of correct responses when someone makes random guesses for 10 multiple choice questions on a SAT test. Each question has 5 possible answers (a,b,c,d,e) one of which is correct. Assume that random guesses are made for each of the 10 questions. Please refer to the table for questions 2-8

X P (x)
0 0.107
1 0.268
2 0.302
3 0.201
4 0.088
5 0.026
6 0.006
7 0.001
8 0+
9 0 +
10 0 +

2) Find the mean number of correct responses?
3) Find the median number of correct response?
4) Find the Standard Deviation for the number of correct responses when random guesses are made for the 10 questions?
5) What is the probability that someone gets at least half of the questions correct?
6) When someone makes guesses for all 10 answers what is the expected number of correct answers?
7) What is the probability of getting a least 1 answer correct/
8) If someone gets at least 1 answer correct does that mean that this person knows something about the subject matter being testee?

9) The television show Grey's Anatomy has a 15% market share when its being broadcast (1.e. 15%of TV sets are tuned into Grey's Anatomy based upon data from Neilsen Media Research) A special focus group consist of 12 randomly selected households (each with one TV set in use during the time Grey's Anatomy broadcast) Apply this to numbers 10 - 14

10) What is the expected number of sets tuned into Greys?
11) In such groups of 12 whish is the mean number of sets tuned into Grey's?
12) In such groups of 12 what is he standard deviation for the number of sets tuned in?
13) For such a group of 12 find the probability that exactly 3 sets are tuned in ?
14) For such a group of 12 would it be unusual to find that no sets are tuned into Greys? Why or why not/

In the year 2001 The New York State Health Department reported a 10% rate of IV virus infections for the "at risk" population in Westchester County - therefore an intensive education program was conducted to lower that 10% rate. After running the program for 5 years a follow up study of 150 "at risk" individuals were surveyed. Apply this to questions 16 and 17

16) Assuming that the program has no effect. Find the mean and standard deviation for the number of HIV cases in the group of 150 people?
17) Among the 150 people in the follow-up study 8% (to 12 people) tested positive for the HIV virus. If the program has effect is the rate unusually low? Does this result suggest that the program is effective?

Inability to get along with others is the reason cited in 17% of employee firings in 2002. Concerned about her company's working conditions the personnel manager at the Westchester Finance Company plans to investigate five employee firings that occurred recently (apply this to questions 19 and 20)

Assuming that the 17% rate applies find the probability that among those five employees the number fired because of an inability to get along with others is at least four.

If the personnel manager actually does find that at least four of the firings are due to an inability to get along with others does this company appear to be very different from other typical companies? Why or Why not?