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HW 3. Due March 23rd, 2015 in class. No late work accepted (this document will post on September 26th, 2014). All of your work should be clearly labeled so there is no confusion on which questions are being answered. Do not hand in an unorganized stack of papers. For the excel exercises, present your results in a reasonable format that would be suitable for a board of directors in a Fortune 500 company. Clearly it is not possible for me to enforce a “no collaboration” policy on this. So, if you decide to work with a classmate(s), just list who you worked with at the top, but hand in your own work. Staple your work before handing in, or it will not be graded. See the syllabus on the partial credit policy. Presentation/professionalism counts for +5 points. In questions 1-7 (numbers 1-4 are binomial, 5-7 are poisson), you will build binomial and poisson probability distributions using excel as we do in lecture. (+10)

 

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1. a basketball player attempts 20 shots from the field during a game. this player hits about 35% of these shots. what is the chance the player hits more than 11 shots? P(X>11)=? (+10)

 

2. Every now and then even a good diamond cutter has a problem and the diamond breaks. for one cutter, the rate of breaks is .1%. If this cutter works on 75 stones, what is the probability that he breaks 2 or more? (+10)

 

3. Ten peas are generated from parents having the green/yellow pair of genes, so there is a .75 probability that an individual pea will have a green pod. Find the probability that among the 10 offspring peas, at least 9 have green pods. Is it unusual to get this result (hint: compare it with the cutoff probability we established for the Rare Event Rule)?

 

4. The television show NBC Sunday Night Football broadcast a game between the Colts and Patriots received a share of 22, meaning that among the TV sets in use, 22% were tuned to the game (based on Nielson data). An advertiser wants to obtain a second opinion by conducting its own survey, and a pilot survey begins with 20 households having TV sets in use at the time of that same NBC Sunday Night Football broadcast. (+3)

a. Find the probability that none of the households are tuned to NBC Sunday Night Football (show with the table in excel and also do using the multiplication rule)? (+3)

b. at least one is (show using the table using the direct method and the complement method)? (+3)

c. exactly one household? (+3) d. find P(X≤3)=? (+10)

 

5. The number of calls received by a car towing service averages 19.2 per day (per 24 hour period). After finding the mean number of calls per hour, find the probability that in a randomly selected hour, the number of calls is 2. (+10)

 

6. On one tropical island, hurricanes happen with a mean of 2.41 per year. Assume the number of hurricanes can be modeled by a Poisson distribution, find the chance that during the next 2 years the number of hurricanes will be 3. (+10)

 

7. Sunita’s job is to provide technical support to computer users. Suppose the arrival of calls can be modeled by a Poisson distribution with a mean of 5.7 calls per hour. What’s the chance that in the next 10 minutes there will be 2 or more calls?

 

8. Here are some practice problems from your text on the Normal Distribution. (+7) Birth weights in Norway are normally distributed with a mean of 3570g and a standard deviation of 500g.

a) If the Ulleval University Hospital in Oslo requires special treatment for newborn babies weighing less than 2700g, what is the percentage of newborn babies requiring special treatment?

b) If the Ulleval University Hospital officials plan to require treatment for the lightest 3% of newborn babies, what birth weight separates those requiring special treatment from those who do not?

 

(+7) Assume that body temperatures are normally distributed with a mean of 98.20 degrees F and a standard deviation of .62 degrees F. Physicians want to select a minimum temperature for requiring further medical tests. What should the temperature be, if we only want 5.0% of healthy people to exceed it?

(+7) Engineers want to design seats in commercial aircraft so that they are wide enough to fit 99% of all males. Men have hip breadths that are normally distributed with a mean of 14.4 inches and a standard deviation of 1.0 inches. Find the hip breadth that bounds the upper 1%.

(+7) The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days.

a) Once classical use of the normal distribution is inspired by a letter to Dear Abby in which a wife claimed to have given birth 308 days after a brief visit from her husband, who was serving in the Navy. Find the probability of a pregnancy lasting 308 days or longer. What does this result suggest?

b) If we stipulate that a baby is premature is the length of the pregnancy is in the lowest 4%, find the length that separates premature babies from those who are not premature. Premature babies often require special care, and this result could be helpful to hospital administration in planning for that care.