Random Sampling: Background You’ll Need 2

Calculating a Sample Mean

In the upcoming course activity, you’ll need to take a sample from a population and ensure that the sampling method you use is unbiased. We’ll learn more about what that really means later. For now, we’ll focus on using the random number generator to select a random sample, then calculate the mean of that sample.

Random Sample

The following table (continued on the next page) displays the number of drivers involved in fatal collisions per billion miles for each of the 50 states.

Ordered Number State Number of Drivers Ordered Number State Number of Drivers
1 Alabama 18.8 26 Montana 21.4
2 Alaska 18.1 27 Nebraska 14.9
3 Arizona 18.6 28 Nevada 14.7
4 Arkansas 22.4 29 New Hampshire 11.6
5 California 12.0 30 New Jersey 11.2
6 Colorado 13.6 31 New Mexico 18.4
7 Connecticut 10.8 32 New York 12.3
8 Delaware 16.2 33 North Carolina 16.8
9 Florida 17.9 34 North Dakota 23.9
10 Georgia 15.6 35 Ohio 14.1
11 Hawaii 17.5 36 Oklahoma 19.9
12 Idaho 15.3 37 Oregon 12.8
13 Illinois 12.8 38 Pennsylvania 18.2
14 Indiana 14.5 39 Rhode Island 11.1
15 Iowa 15.7 40 South Carolina 23.9
16 Kansas 17.8 41 South Dakota 19.4
17 Kentucky 21.4 42 Tennessee 19.5
18 Louisiana 20.5 43 Texas 19.4
19 Maine 15.1 44 Utah 11.3
20 Maryland 12.5 45 Vermont 13.6
21 Massachusetts 8.2 46 Virginia 12.7
22 Michigan 14.1 47 Washington 10.6
23 Minnesota 9.6 48 West Virginia 23.8
24 Mississippi 17.6 49 Wisconsin 13.8
25 Missouri 16.1 50 Wyoming 17.4

We would like to select a random sample of six states from this list. We will select this sample “without replacement,” meaning that we cannot select the same state twice. The word random, used statistically, means that each individual in the population has the same chance of being selected in the sample.

Question 2

Describe how you could use cards to select a random sample of six states, without replacement. In your description, include the number of cards you would need, what you would write on each card, and how you would select your sample.

Now, instead of using cards, you are going to use a random number generator to select a random sample of six states. You’ll use the tool to do this. If you work with a partner, use the sample you generate together in Question 3 to answer Question 4.

question 3

Go to the Generate Random Numbers tool at https://dcmathpathways.shinyapps.io/RandomNumbers/.

Step 1) Select the Random Numbers tab.

Step 2) Under “Choose Minimum,” select “1.”

Step 3) Under “Choose Maximum,” select “50.”

Step 4) Under “How many numbers do you want to generate,” select “6.”

Step 5) Under “Sample with Replacement,” select “No.”

Step 6) Click “Generate.” This will generate six random numbers between 1 and 50. These six numbers correspond to the states chosen for your sample (locate the number next to its corresponding state name in the data list above). Fill in the following table with the corresponding state and number of drivers involved in fatal collisions per billion miles for each of your randomly-generated numbers.

Randomly

Generated Number

 State Number of Drivers

Involved in Fatal Collisions

per Billion Miles

Sample Mean

To understand what the typical number of drivers involved in fatal collisions might be, we can calculate the mean or average of the values. The mean is calculated by adding the values and then dividing the total by the number of values in the dataset. When we calculate the mean of a random sample, we call this the sample mean.

Recall

Core skill:

See the example below, then answer Question 4 using the random sample you generated from the data table during Question 3.

example

Suppose a random sample of states was taken from the list above: Minnesota, Nevada, West Virginia, Tennessee, Alaska, Indiana.

To calculate the mean number of drivers involved in fatal collisions per billion miles for this sample, we take the sum of the number of drivers per state in the sample, then divide by the total number of states in sample.

  1. What are the number of drivers involved in fatal collisions per billions associated with each state in the sample? Look these up in the data table.

  2. Calculate the sample mean.

Now you try it.

question 4

Calculate the sample mean (i.e., average) number of drivers involved in fatal collisions per billion miles for the six states in your randomly-selected sample.

question 5

If you were to use the random number generator to generate another simple random sample of six states, would you get the same six numbers you found in Question 3? Would you get the same value for the sample mean number of drivers involved in fatal collisions per billion miles you found in Question 4? Explain.

You’ve had an introduction to the analysis tool, learned about dotplots, had a chance to refresh your skills at computing the mean, and learned how to use the random number generator in order to choose a random sample from a dataset. Hopefully, you are feeling secure with these new abilities and are ready to move on to the next section and activity.