Name(s)                                                                 :
Sampling Distributions and Variability
Introduction to Statistics, Spring 2001, Tom Linton and Wendy Weber
See the change! See the class data.
Work in groups of size two.
In today's activity, we will look at the shapes of the distributions of various statistics associated with samples. We want to look at how the shape of the distribution of  values behaves, as we change the sample size, n, used to calculate . To get started, we need to make up a population of numbers using M&Ms!
  1. Take your collection of M&Ms and randomly divide them into 2 piles of about the same size. You need NOT be exact, just divide them roughly into 2 similar sized piles.

    2.  
  2. In each pile, let
      3. T = the total number of M&Ms,
      B = the number of brown M&Ms,
      R = the number of red M&Ms,
      Y = the number of yellow M&Ms, and
      O = the number of M&Ms that are green, orange or blue.
    Record the numbers for each pile above, on the appropriate lines.
     
  3. Each of your two piles of M&Ms will be used to create three different individuals for our population. All of the individuals will turn out to be numbers that are approximately equal (typical values will be from 0.5 to 1.5). The first individual will be the number B / (R + Y) rounded to two decimal places. Record these values for each group below.

    4.  

     
     
     
     
     

  4. The second individual will be the number (3*R) / (T - R) rounded to two decimal places. Record these numbers for each group of M&Ms below.

    5.  

     
     
     
     
     
     

  5. The third number will be the number (2.5*O) / T rounded to two decimal places. Record these numbers for each group of M&Ms below.

    6.  

     
     
     
     
     
     

  6. Eat your M&Ms!

    7.  

     
     
     

  7. Add all 6 of your individuals to the population data on the board. Copy down the class data as a stem plot, with split stems, and be sure to put the data in order (from smallest to largest) on your stem plot.

    8.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

  8. Calculate the mean and standard deviation of the population (the collection of all individuals from the entire class). Record these below.

    9.  

     
     
     
     
     
     

  9. Are the two numbers from (8) statistics or parameters? Also record N = the number of individuals in the population.

    10.  

     
     
     
     
     
     

    We will now draw several samples from our population and calculate  for each sample. To do this, we should label the individuals in our population with numbers from 1 to N, and then use our calculator's randInt command (do NOT seed your calculator however). We would use randInt(1,N,4) to select 4 numbers and randInt(1,N,10) if we wanted 10 individuals in our sample. Of course, we should remove duplicates in our samples with repeated calls to the command randInt(1,N). Once we have a sample of the correct size, we take those individuals from the population and calculate their average. For example, if your randInt command returns the values 3, 12, 9, 2, then you will use the 3rd smallest individual, the 12th smallest individual, and so on, for your calculation. It is probably easiest to calculate the  values by hand (something like (.86 + .43 + .98 + .34) / 4 ), rather than using 1VarStats.
     
     
     
     

  10. Select 5 SRS's of size 4 from the population and calculate their averages. Record the information below and record your  values on the board (rounded to two decimal places).
Samples of size 4
Sample SRS labels      Individuals  average
1      
2
      
3
      
4
      
5
      
  1. Did every group get the same  values in their samples?

    2.  

     
     
     

  2. Record the class data below with a stem plot. Is this data more or less spread out than the population?

    3.  

     
     
     
     
     
     
     
     
     
     
     
     
     

  3. Does the stem plot look fairly symmetric?

    4.  

     
     
     

  4. Is the center of this data set about the same as the center of the population?

    5.  

     
     
     
     

  5. Calculate the mean and standard deviation of the class's collection of  values from samples of size 4. The text predicts that the mean of these  values will be the same as the mean of the population. Does this seem to be correct here?

    6.  

     
     
     
     
     

  6. The text also predicts that the standard deviation of the class's  values will be about half of the standard deviation of the population. Does this seem to be correct here?

    7.  

     
     
     
     

  7. Select 5 SRS's of size 16 from the population and calculate their averages. Record the information below and record your  values on the board (rounded to two decimal places).

    8.  
    Samples of size 16
    Sample      Individuals  average
    1    
    2
    		    
    3
    		    
    4
    		    
    5
    		    
  8. Record the class data below with a stem plot. Is this data more or less spread out than the population? Is it more or less spread out than the samples of size 4 data?

    9.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

  9. Did every group get the same  values in their samples?

    10.  

     
     
     
     
     

  10. Is this data set fairly symmetric? Is it more or less symmetric than the samples of size 4? How about the original population?

    11.  

     
     
     
     
     
     
     

  11. Is the center of this data set about the same as the center of the population?

    12.  

     
     
     
     
     
     
     

  12. Calculate the mean and standard deviation of the class's collection of  values from samples of size 16. The text predicts that the mean of these  values will be the same as the mean of the population. Does this seem to be correct here?

    13.  

     
     
     
     
     
     
     

  13. The text also predicts that the standard deviation of the class's  values will be about one fourth the standard deviation of the population. Does this seem to be correct here?

    14.  

     
     
     
     

  14. We've seen stem plots and numerical summaries of the population, a collection of  values from samples of size 4, and a collection of  values from samples of size 16. Summarize what you've seen about the centers of these three distributions (are they all the same, are some larger, smaller?).

    15.  

     
     
     
     
     
     

  15. Likewise summarize what you've seen about the standard deviations of these three distributions.