Name(s)                                                                  :
Sampling Distributions and Variability
Introduction to Statistics, Fall 2000, Tom Linton and Wendy Weber
Work in groups of size two.
  1. Take your collection of M&Ms and count out the "first" 25 of them. Divide (randomly) the remaining M&Ms into groups of size 10. Once you have less than 10 remaining, you may eat the leftovers (the last group of less than 10).
  2. We are interested in estimating two quantities, the percentage of M&Ms that are yellow and the percentage of M&Ms that are brown. To accomplish this, we will calculate the proportion of yellow M&Ms in our piles of size 25 and the proportion of brown M&Ms in each pile of size 10.
    1. What two parameters are involved in this process?

      2.  

       
       
       
       
       
       
       
       
       

    2. Describe the two statistics we will use to estimate the parameters in part (a). Note: you should be able to describe a statistic before you calculate its value.

      3.  

       
       
       
       
       
       
       
       
       
       
       

  3. Record the proportion of yellow M&Ms in your group of size 25.                             :

    4.  

     
     
     
     
     
     
     
     
     

  4. In the table below, record the proportion of brown M&Ms in each group of size 10.
Brown M&M Proportions
Group 1 2 3 4 5 6
Proportion        
  1. Add your proportions to the class data plots and then copy these plots below.

    2.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

  2. Did every group get the same proportions in their samples?

    3.  

     
     
     
     
     

  3. Do the data plots exhibit a recognizable shape, or do they look like random collections of numbers?

    4.  

     
     
     
     
     

  4. Are the data plots roughly symmetric? If so, guess the values of the center of each distribution?

    5.  

     
     
     
     
     
     

  5. Do they appear approximately bell-shaped?

    6.  

     
     
     
     
     
     
     
     
     
     
     

  6. In fact, the class proportions are suppose to exhibit very predictable behavior. Obviously, when I wrote this question, I had no idea what proportions the class would obtain. Nonetheless, I'm going to make four predictions about the classes' data. For each of my predictions, you should calculate what percentage of the class data actually satisfies the inequalities in the prediction.
    1. I predict that roughly 40% of the class proportions of yellow M&Ms will be less than 0.18.

      2. Actual percentage =




       

    2. I predict that roughly 15% of the class proportions for brown M&Ms will be larger than 0.45.

      3. Actual percentage =




       
       

    3. I predict that roughly 50% of the yellow M&M proportions will be between 0.15 and 0.25.

      4. Actual percentage =




       
       

    4. I predict that roughly 65% of the brown M&M proportions will be less than 0.35.

      5. Actual percentage =




     

  7. Calculate the mean and standard deviation of both the proportions of brown and yellow M&Ms based on the class data sets. My predictions from above assumed that means were yellow = 0.2, brown = 0.3, and the standard deviations were yellow = 0.09, brown = 0.15. Were my predicted means and standard deviations close?

    8.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

  8. Using your calculator's randInt command (do NOT seed your calculator with the problem number though), select an SRS of size 4 from the class data set of proportions of yellow M&Ms. To do this, number the yellow proportions 1, 2, 3, ..., n, where n is the total number of yellow proportions in the class data set (arrange the yellow proportions from smallest to largest). Use randInt(1,n,4) to select 4 numbers and then calculate the average of the 4 yellow proportions that correspond to your sample of size 4. For example, if your randInt command returns the values 3, 12, 9, 2, then you will use the 3rd smallest yellow proportion, the 12th smallest yellow proportion, and so on, for your calculation. Record the yellow proportions you selected and the mean of your 4 yellow proportions below. Add your 4-proportion mean to the class data plot on the board and then copy the class data set below.

    9.  

     
     
     
     
     
     
     

    yellow proportion numbers in our SRS:

    mean of our 4 selected yellow proportions =
     
     
     
     
     
     
     
     
     
     
     
     
     

  9. Calculate the mean and standard deviation of this new data set.

    10.  

     
     
     
     
     
     
     
     
     
     
     
     

  10. Is the new mean approximately equal to the mean for yellow proportions from question 11?

    11.  

     
     
     
     
     
     
     
     
     
     

  11. Is this new standard deviation about half of the yellow standard deviation from question 11?