## Estimating the Mean State Area

The table below gives the areas (in thousands of square miles) for each of the “lower 48” states. This serves as the population for this study. Your task involves taking small samples from this population and using the sample mean to estimate the mean area for the population of states by following the steps indicated below.

Your challenge is to discover some important properties of random samples, properties that illustrate why random sampling is the key to getting good statistical information about a population. In this task, unlike “real life” situations, you will have all of the population data at hand, and will use it to see how random sampling works. Your classmates will have the same task, and you will be combining your data with theirs. In the first part of the task, you will select a sample of states by a method of your choice. For the second part you will follow a specified procedure.

**Procedure #1: Choose your own sample**

- By any quick method you like, select 5 states that you think represent the 48 (perhaps by tossing 5 grains of sand on a map of the states and selecting the states on which they fall; shutting your eyes and pointing your finger at a spot on the map, repeating the process until 5 states are selected; systematically selecting a state from the northeast, the south, the mid-west, all east of the Mississippi, and two states from west of the Mississippi).
- Find the areas of these 5 states and calculate the mean for your sample.
- As a class, construct a dot plot of the sample means.

**Procedure #2: Use random sampling**

- Number the 48 states from 1 to 48. Then use a random number table or a random number generator to obtain 5 random numbers between 1 and 48, and then find the states corresponding to these numbers.
- Find the areas of these 5 states and calculate the mean for your random sample.
- As a class, construct a dot plot of the sample means from the random samples.
- Compare the plots produced in steps c and f. Where are the centers? Which has greater spread?
- Repeat steps a - f for random samples of size 10 and compare the plots. What differences, if any, do you see in the plots? What feature or features appear to stay the same?
- Find the actual mean state area using the data from all the states. Summarize at least two important points concerning the value of random sampling.

State | Area |
---|---|

Texas | 269 |

California | 164 |

Montana | 147 |

New Mexico | 122 |

Arizona | 114 |

Nevada | 111 |

Colorado | 104 |

Oregon | 98 |

Wyoming | 98 |

Michigan | 97 |

Minnesota | 87 |

Utah | 85 |

Idaho | 84 |

Kansas | 82 |

Nebraska | 77 |

South Dakota | 77 |

Washington | 71 |

North Dakota | 71 |

Oklahoma | 70 |

Missouri | 70 |

Florida | 66 |

Wisconsin | 65 |

Georgia | 59 |

Illinois | 58 |

Iowa | 56 |

New York | 55 |

North Carolina | 54 |

Arkansas | 53 |

Alabama | 52 |

Louisiana | 52 |

Mississippi | 48 |

Pennsylvania | 46 |

Ohio | 45 |

Virginia | 43 |

Tennessee | 42 |

Kentucky | 40 |

Indiana | 36 |

Maine | 35 |

South Carolina | 32 |

West Virginia | 24 |

Maryland | 12 |

Massachusetts | 11 |

Vermont | 10 |

New Hampshire | 9 |

New Jersey | 9 |

Connecticut | 6 |

Delaware | 2 |

Rhode Island | 2 |