Activity 20.1 -- MODIFIED AND SHORTENED!!!!   : )

Sampling and Surveys

10 points

Due at 12:00 noon, Thursday, April 30, 2009

PLEASE ANSWER ALL THE QUESTIONS GIVEN HERE.  THEY HAVE BEEN MODIFIED APPROPRIATELY.

In this activity, you will investigate why random sampling is important and also consider some issues involved in the design of surveys.

1. Here is a list of the 50 United States.

1. a. Using your sense of the land area of each state, choose what you think is a representative (that is, your subjective) sample of six states. Your goal is to pick a sample of six states whose mean land area is close to the mean land area of all 50 states.(It might help to try to visualize a map of the U.S. to get your sample of six "representative" states.)

Copy and paste this table into your Word document. It should be filled with the six states you choose. AFTER you choose the states, you should look up the land area of each state, in square miles, and enter them in the table as well. You may use the Information Please Almanac for the land areas.

1. b. Compute the sample mean for the land area of your subjective sample of six states, and enter it into your Word document.

1. c. Now use Excel's random number generator to generate six random integers between 0 and 49. Recall how you generated random integers 0, 1, 2, and 3 in Activity 18.1 (= INT( 4 * RAND( ) ) )and adapt that technique to generate integers between 0 and 49. It is possible that you might get repeated numbers when you generate the random integers. If you do, discard the second occurrence of the repeated number and generate another integer so you have six distinct integers.

Hint:  Since Excel recalculates random numbers whenever you do anything, you might want to copy and "paste values" into a different column to "freeze" your random numbers.

Enter these six random integers into your Word document, and briefly explain the process you used to generate them.

1. d. Number the states in a systematic way, and use the numbers you generated in #1.c. of this question to pick a random sample of six states. Explain your systematic method for numbering the states.

1. e. Copy and paste the table below into your Word document. It should be filled with the six states you randomly chose, and their land areas, in square miles. Once again, you may use the Information Please Almanac for the land area data.

1. f. Compute the sample mean for the land area of your random sample of six states, and enter it into your Word document.

1. g. Recall that a population refers to the whole group about which you want to draw a conclusion, and a sample refers to a subgroup of the population. The population mean land area for the population of all 50 states is approximately 70,748 square miles.  Compare your sample means from #1.b. and #1.f. to the population mean given here.  Comment on the differences among these three means.  Were these differences expected?

2. Use Excel to generate a stratified random sample of six states, stratified by location ("East of" or "West of") relative to the Mississippi River.

From Topic 20, Exploration #9, here is a list of states that are east of the Mississippi River:
Alabama, Connecticut, Delaware, Florida, Georgia, Illinois, Indiana, Kentucky, Maine, Maryland, Massachusetts, Michigan, Mississippi, New Hampshire, New Jersey, New York, North Carolina, Ohio, Pennsylvania, Rhode Island, South Carolina, Tennessee, Vermont, Virginia, West Virginia, Wisconsin.

Since this is a stratified sample, you will first need to determine the size of each stratum. In particular you'll need to count the number of states east of the Mississippi, and the number of states west of the Mississippi. Then, the proportion of states east of the Mississippi in your sample size of six should be the same proportion as the total number of states east of the Mississippi. Also, be sure to choose the proper number of states from each stratum (East and West), or you won't be able to use the state numbering used above in #1. d.

Briefly describe how you generated your stratified random sample. Your description should include how you numbered the states in each stratum, and the random numbers you generated for each stratum.

2. a. Copy and paste the table below into your Word document. It should be filled with the six states of your stratified random sample, and their water areas, in square miles. Once again, you may use the Information Please Almanac for the water area data.

2. b. Compute the sample mean water area for your stratified random sample of six states and record it in your Word document.

2. c. Describe how your sample mean water area compares to the population mean water area of approximately 5,133 square miles.

2. d. Explain why stratifiying a sample, as we just did in #2, is useful.

Summary
In this activity, you investigated why you would want to collect random samples rather than "subjective" samples. You also collected a stratified random sample. To collect these random samples, you used Excel's random number generator.