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Lab 3 – Binomial Distribution

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					Lab 2 – Probability and Modeling Discrete Random Variables




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PART A -DISCRETE RANDOM VARIABLES – BINOMIAL AND POISSON

In a recent CBS poll, 72% of American people support a public option for health care insurance. Suppose you
take a random sample of 40 American adults. Let X be the number in the sample who support a public option for
health care insurance.

   1. Model X as a binomial random variable. Use The MEGASTAT>PROBABILITY>BINOMIAL
      DISTRIBUTION function to build a table in Excel for individual and cumulative probabilities.




   2. Describe the shape of the Column graph of the distribution of individual probabilities.




   3. Determine the probability that X is between 25 and 30 inclusive.



   4. Suppose 25 people in the sample support a public option for health care insurance. Would that be
      unusual? (Find the probability X≤25)



   5. Suppose 34 people in the sample support a public option for health care insurance. Would that be
      unusual? (Find the probability X≥34).




   6. Calculate the population mean and standard deviation for this random variable.
Use the MEGASTAT>PROBABILITY>POISSON DISTRIBUTION function to build a table in Excel for individual
and cumulative probabilities.

   Strong earthquakes (of RM 5 or greater) occur on a fault at a Poisson rate of 1.23 per year.

   7. Find the probability distribution function for the number of strong earthquakes over one year. Create a
      table from MEGASTAT.




   8. Find the probability distribution function for the number of strong earthquakes over 15 years. What is the
      new value for the rate? Create a table from MEGASTAT.




   9. Compare the shape of the 2 graphs.


   10. From these tables, find the following probabilities:

           a. Exactly 2 Strong earthquakes in the next year.



           b. At least 2 Strong earthquakes in the next year.



           c.   At most 2 Strong earthquakes in the next year.



           d. Between 8 and 17 (inclusive) Strong earthquakes in the next 15 years.



           e. No Strong earthquakes in the next 2 years. (Don’t create a new table to solve this problem.
              Instead, assume that years are independent and use the multiplicative rule for independent
              events.)
Part B Modeling Continuous Random Variables – Simulation

This lab covers simulation of Continuous Random Variables using the Random Number Generator in Excel (the
RAND() function) and the Inverse functions of continuous distributions we have covered in the class.
The RAND() functions returns a random probability between 0 and 1. This probability is then converted to a
percentile by the appropriate function:

   Formula For Uniform(a,b): let the cell be: = (b-a)*RAND()+a where a and b are the two endpoints.
   Formula For Normal(): let the cell be: = NORMINV(RAND(),,).where  and  are the mean and
    standard deviation.
   Formula For Exponential(): let the cell be: = - LN(1-RAND()).where  is the mean.

Steps for each problem 1-3 below
    i. Simulate approximately 1000 trials.
    ii. Calculate the sample mean, median and standard deviation. Use the functions AVERAGE , MEDIAN,
         STDEV, MIN and MAX. Compare to the population mean, median and standard deviation. (This is
         called verification of the model.)
    iii. Using earlier techniques, make a histogram and box plot and comment on the shape.
    iv. Paste in the summary statistics, graphs and your observations only. DO NOT PASTE IN SIMULATED
         RAW DATA.

To help you get started, the worksheet simulations.xls has complete simulations for part a of each problem as
well as calculation of the sample statistics. You still need to make graphs and add analysis for parts 1a and 2a
in addition to completely answering parts 2b, 2c and 3.

    1. Uniform Distribution

            a. The waiting time for a scheduled bus follows a uniform distribution from 5 to 40 minutes.


            b. The amount of chemical sold by a contractor follows a uniform distribution from 10 to 55 metric
               tons.


    2. Normal Distribution

            a. The height of men follows a Normal Distribution with =70 inches and =4 inches.


            b. The height of women follows a Normal Distribution with =67 inches and =3.5 inches.


    3. Exponential Distribution

            a.   Earthquakes of RM 5 or greater occur on a fault with an expected waiting time of 1.5 years. Let X
                 be the waiting time until the next earthquake (which follows exponential distribution.)

            b.   Trauma patients arrive at a hospital’s Emergency Room at a rate of every 10 minutes. Let X be
                 the waiting time until the next patient arrives (which follows exponential distribution.)

				
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