In this lab, we will be simulating various sampling distributions for sample means.  We will start out looking at a normal population then at some non-normal populations.  In doing so, we will investigate how the sampling distributions of the mean is affected by sample size and its population's distribution.

1. Suppose the weights of Mounds fun size candy bars are normally distributed with a mean of 20.30 grams and a standard deviation of 0.55 grams.  We are going to have Minitab simulate the weights of 1000 of these candy bars as well as simulate the mean weights of 1000 samples of size 10.  We will then compare the means, standard deviations, and shapes of these two distributions.
1. Have Minitab simulate 1000 weights of Mounds candy bars and store the 1000 values in the column c1. 
• To do this, click on Calc > Random Data > Normal. 
• In the window that opens up, click on Generate … rows of data box and type in 1000.  Then click in the Store in columns box and type c1, click in the Mean box and type 20.30, click in the Standard deviation box and type in 0.55, and click OK. 
  
• Construct a histogram of your data such that the horizontal axis goes from 18 to 23 in increments of 0.1. 
• To do this, click on Graph > Histogram > Simple > OK.  Double click on the horizontal axis of your histogram to open up the Edit Scale window.  Click on the Binning tab, click on cutpoint, the click on Midpoint/cutpoint positions and enter 18:23/0.1.   Label the horizontal axis of the histogram, "Individual Candy Bar Weights." 
  
• Find the mean and standard deviation for the data.
• Stat > Basic Statistics > Display Discriptive Statistics
  

2. We now want to simulate drawing 1000 samples of size 10. 
   
• To do this, repeat what you did in question 1(a), but place your simulated data in columns 3 through 12. Think of the entries in row 1 of columns 3-12 as the first sample of size 10, the entries in row 2 of columns 3-12 as the second sample of size 10, etc.  Click Calc > Random Data > Normal as before.  At the Normal Distribution window do everything as before, except now in the Store in columns box type c3-c12 then click OK.
• You should now have 1000 rows of data in each of columns 3 through 12.  For each row, we want to compute the sample mean for the 10 values in columns 3-12 of that row and place the sample mean in the correct row of column 13.
  
• To do this, click Calc > Row Statistics. In the Row Statistics window click Mean, in the Input variables box type c3-c12, and in the Store result box type in c13, and click OK. 
  
• The means of your 1000 repetitions of samples of size 10 should now be in column 13.  Construct a histogram of your sample means using the same range of values on the horizontal axis as your graph from question 1(a).  Label the horizontal axis of the histogram, "Sample Mean Weights of Candy Bars."

3. What are the similarities and differences between the histogram from question 1(a) and that from question 1(b)?
4. Find the mean and standard deviation for the mean data.  How do these values compare to the mean and standard deviation of the original individual candy bar weights from question 1(a)?

  

2. In the previous questions the original population had a normal distribution.  In this question we see a non-normal population and will thus see the Central Limit Theorem in action. Open up a new Minitab worksheet (File > New > double click on Minitab Worksheet.)  Put the penny data in this new worksheet.  This data set gives the years of 204 pennies that were collected by your professor.
1. Construct a histogram of the penny data such that the horizontal axis goes from 1960 to 2002 in increments of 2.    Make sure you label your graph properly.  Find the mean and standard deviation for the data.
2. We now want to simulate drawing 1000 samples of size 10 as was done in question 1.  You should again put your data in columns 3-12. 
   
• To do this Click Calc > Random Data > Sample from Columns.  Sample 1000 rows from the column pennies, and store samples in column c3.  Make sure you click in the sample with replacement box.  Repeat this for columns 4 - 12.  You should now have 1000 rows of data in each of columns 3 through 12. For each row, we want to compute the sample mean as in question 1.  Again store these in column 13.
• Construct a histogram of your sample means using the same range of values as your graph from question 2(a).  Label the horizontal axis of the histogram properly.  What are the similarities and differences between this histogram and that from question 2(a)?
3. Find the mean and standard deviation for your mean date data.  How do these compare with the mean and standard deviation for the original penny data?  Theoretically, approximately what are the mean and standard deviation for a sampling distribution of size 10 from the penny data?

3. We will now see how a sampling distribution for the mean changes as you increase the sample size.  To do this, go to the Rice Virtual Lab in Statistics.  Select "Begin" under "Sampling Distributions" on the left side of the page to answer the following questions.  In the parent population (the top graph) select skewed.  Now generate an approximate sampling distribution for n = 2.  To do this, go to the boxes to the right of the third graph from the top and select Mean, N=2, and click on Fit Normal.  Now, click on 10,000 Samples and you should see an approximate sampling distribution.
1. How does the shape of your sampling distribution compare with that of the parent population and that of a normal distribution?
2. How does the mean of your sampling distribution compare with that of the parent population?
3. How does the standard deviation of your sampling distribution compare with that of the standard deviation of your parent population divided by the square root of the sample size?
4. Repeat this process for N=5 and answer parts a, b, and c for this new sampling distribution.
5. Repeat this process for N=10 and answer parts a, b, and c for this new sampling distribution.