Rectangularity:  Part I.  Sampling Distribution of the Sample Mean

 

 

Purpose:

This activity is intended to illustrate properties of the sampling distribution of a sample mean.

 

The Population of Rectangles Sheet shows a population of size 100 consisting of rectangles of varying areas.  Each square counts as one unit towards a rectangle’s area.  The true average (mean) area of the rectangles in the population is m=6.26.  The true standard deviation of the areas of the rectangles in the population is s=5.69.  If we did not know m and wished to estimate it, we could draw a simple random sample of rectangles from the population and use the mean area of the sampled rectangles to estimate m.  The sample mean,will vary from sample to sample.  The distribution of the values for many simple random samples of size n is called the sampling distribution of the statistic .  

 

 

Instructions:

Work in groups of three.  Each group should have a random number table or a calculator capable of generating random numbers, a copy of the activity worksheet, and a copy of the questions sheet.

 

Label the rectangles in the population from 00 to 99.  (Call Rectangle 1, 01; call Rectangle 2, 02; and so on up to Rectangle 99, which you should call 99.  Call Rectangle 100, 00).

 

1.  Select two different simple random samples of size 5 from the population (sample with replacement -- so that it is possible to select the same rectangle more than once).  For each sample, list the labels of the rectangles selected, list the areas, and then calculate the value of . Complete the tables below.    After you have completed the tables, write your two values for  on the whiteboard under Sample Size n=5.  Once the entire class has finished with random samples of size n=5, complete the n=5 column on the data collection sheet.  (The data collection sheet is given at the end of Part I.)

 

            Random Sample 1                                                                                Random Sample 2

 

               =                                                                                                      =

 

2.  Select two different simple random samples of size 15 from the population (sample with replacement).  For each sample, list the labels of the rectangles selected, list the areas, and then calculate the value of .  Complete the tables below.  After you have completed the tables, write your two values for  on the whiteboard under Sample Size n=15.  Once the entire class has finished with random samples of size n=15, complete the n=15 column on the data collection sheet.

 

 

 

             Random Sample 1                                                                               Random Sample 2

 

                          =                                                                                                     =  

 

 

3.  Select two different simple random samples of size 25 from the population (sample with replacement).  For each sample, list the labels of the rectangles selected, list the areas, and then calculate the value of .  Complete the tables below.  After you have completed the tables, write your two values for  on the whiteboard under Sample Size n=25.  Once the entire class has finished with random samples of size n=25, complete the n=25 column on the data collection sheet.

 

            Random Sample 1                                                                                Random Sample 2

 

                           =                                                                                                     =

 

 

 

 

Questions:

 

Answer the following questions using the data table on the data collection sheet.

 

1.  For each sample size n = 5, 15, and 25 construct a histogram of the sample mean values.

 

 

 

2.  For each sample size, describe the shape of the distribution of  values. 

 

 

 

3.  Compare the shape of the distributions of the  values to the shape of the distribution of the population.  Which looks more normal?

 

 

 

4.  Based on your histograms, what do you think is the relationship between the sample size and the shape of the distribution of the sample mean?

 

 

 

5.  (a) For each sample size, calculate the standard deviation and the mean of the sample means. 

 

 

 

(b) For which sample size is the standard deviation the largest and for which sample size is the standard deviation the smallest?  Why do you suppose this happens?

 

 

 

6.  How does the standard deviation of the  values compare to the standard deviation of the population?  What does this tell you about the spread of the  values compared to the spread of the population values?

 

 

 

7.  Find an expression for the mean of the sample means,as a function of the mean of the population, m.

 

 

 

8.  Try to develop a formula to relate the standard deviation of the sample means,to the population standard deviation, s, and the sample size, n.  (Hint:  the formula involves)

Data Collection Sheet:

 

                                           Data Table.  Class Sample Means