Name(s)                                                   :
Random Variables That Count Successes
Math 203, Introduction to Statistics,
Spring 2001, Tom Linton, Central College
Class Data
The following types of situations arise 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 an experiment or trial 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 = 0.333.

Let's simulate this situation. We want to repeat an experiment N = 72 times where each trial, or run of the experiment results in a success (finding your partner) 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. If, for example the command randBin(72, 0.333) produces the number 25, it means that 25 out of the 72 couples correctly selected their partner. This command is located on the [MATH] [PRB] menu in location [7:RandBin].
 

  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.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

  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), the probability that the random variable X is less than or equal to 20..

    3.  

     
     
     
     
     
     
     
     
     

  3. Estimate P(X >= 30).

    4.  

     
     
     
     
     
     
     
     

  4. Estimate P(X = 24).

    5.  

     
     
     
     
     
     
     

  5. Theory predicts that a binomial random variable with N = 72 and p = 0.333 will take on the value X = 24 with probability 0.099. Is this value close to your answer to the last question?

    6.  

     
     
     
     
     
     
     

  6. You can calculate binomial probabilities with your TI-83 calculator. The command binompdf(N,p,X) gives the probability that a binomial random variable with N trials, and probability of success equal to p takes on the value X (a whole number from 0 to N). For example, I got the value 0.099 in the last question by entering binompdf(72, .333, 24). The binompdf command is located on the [DISTR] menu in location number 0 (zero, or 10). Use your calculator to calculate the probability that a binomial random variable with N = 72 and p = 0.333 takes on the value X = 25.

    7.  

     
     
     
     
     
     
     

  7. Use the class data to estimate P(X = 25). Is this value close to your answer from the last question?

    8.  

     
     
     
     
     
     
     

  8. Note that the article gave results as percents, 58% and 69% of the couples were successful in picking their lover. Find the values of X (they must be integers) that correspond to 58% and 69% of 72. Call these two X values A (for 58%) and B (for 69%). What proportion of the class data had X > A? How about X > B?

    9.  

     
     
     
     
     

  9. Do you think love is blind? That is, do you think the values in the article are similar to the values in the class data set?

    10.  

     
     
     
     
     
     
     
     
     

  10. Your calculator can also calculate binomial probabilities associated with certain types of intervals, namely things like P( X <= k) (the probability that X is less than or equal to k). The command binomcdf(N, p, k) will calculate the probability that X = 0, 1, 2, ... or k (i.e. that X is less than or equal to k). Use your calculator to calculate the probability that a binomial random variable with N = 72 and p = .333 takes on a value that is less than A (from the last question). This is the same as being less than or equal to A - 1.

    11.  

     
     
     
     
     

  11. Repeat the last question for k = B - 1.





  12. Now realize that since X must take some value from 0 to 72, and all probabilities sum to 1, the probability that X >= A is just 1 minus the probability that X is less than or equal to A - 1. Find P(X >= A).






  13. Repeat the last question for P(X >= B).

    14.  

     
     
     
     
     

  14. Why do your  last 4 answers also indicate that love is not blind?

    15.  

     
     
     
     
     
     
     
     

  15. 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 (the class data mean and the number N*p) close?

    16.  

     
     
     
     
     
     
     
     
     

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