Name(s)                                                    :
Random Variables That Count Successes
Math 203, Introduction to Statistics, Fall 2000, Tom Linton, Central College
The following situations arise quite frequently:

Select a random sample of size 15 from a large population where 60%,
or p = 0.6, of the population has a property (like believes milk costs too much).
Define a random variable X to be equal to
the number out of 15 that believe milk costs too much.

OR

Repeat a simple experiment a fixed number of times, say N = 24 times.
Designate some event, say A, for this experiment to be a success.
Define a random variable X to be the number of trials (out of N)
of your experiment that end with a success.

The purpose of today's activity is to simulate outcomes from this type of situation, estimate probabilities using our simulation and calculate statistics based on our simulation. Random variables like those above are called Binomial Random Variables.

Read the handout "Love is not blind, and study finds it touching".

We can define our experiment to be one partner trying to locate their lover, amongst three persons. A success in our experiment is defined as one lover correctly locating their partner. We repeat this experiment N = 72 times. Our trials (repetitions of the experiment) are independent, since one partner finding (or missing) their lover has no effect on the next couple. If the partners simply guess which of the three persons is their mate, they would be correct p = 1 / 3 of the time, or about p = 0.3333. Let's simulate this situation. We want to repeat an experiment N = 72 times where each trial results in a success with probability p = 0.333. Let's define X to be the number, out of 72, that properly select their partner, and collect a large number of simulated values of the variable X. The command randBin(72, 0.333) will (after a while) spit out a single simulated value of the variable X.
 

  1. Run the randBin command above 12 times and record your values below.
randBin(72, 0.333) values
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12
                       










     
  1. Add your values to the class data set on the board and make a histogram of the class data set below.















  2. What proportion of the class data set had X values less than or equal to 20? This is an estimate of P(X <= 20).





  3. Estimate P(X >= 30).




  4. Estimate P(X = 24).




  5. Estimate P(X = 25).





  6. What proportion of the class data had X > 58? How about X > 69? Do you think love is blind?








  7. Calculate the mean of the class data set. Note: There is a shortcut to doing this. If you enter the X values in L1 and the counts in L2 and issue the command 1-VarStats L1, L2  the value of x-bar returned is the correct mean. Compare your mean to the value N*p (N = 72 and p = 0.333). Probability theory states that N*p is the mean for situations like this. Are the two values close?






  8. Calculate the standard deviation of the class data set (it should have been given by the command above). Compare this value to the theoretical value ( N*p*(1 - p) )^0.5. Are these two numbers close?