The probability of an event can be represented by using an area model. The diagram below depicts the outcomes of flipping a coin. Blue represents heads, and red represents tails. The probabilities for getting heads and tails are equal. Therefore, each outcome is represented by an equal area in the diagram. This type of graphical representation is useful for comparing and estimating probabilities visually. Geometric models are often easier to use because they require fewer computations after the visual model is created.

Cartograms are a type of map that can be used to represent geographic population data. Regions within the cartogram are scaled according to the value of some variable that varies by region. Well-made cartograms preserve some aspect of the land area’s original shape, which helps viewers to understand the diagram more easily. For this reason, cartograms are almost always made by using computer software.

Cartograms can help to visualize the probabilities of certain events, such as the probability of an American citizen being born in a given state. They can also be useful for making inferences, or conclusions, from data. In this part of the assignment, you will use cartograms to make inferences about how and why populations change.

There are three main reasons for population change: the number of births, number of deaths, and amount of migration. Migration is when people move into and out of a region. Immigration is when people move into a region. Emigration is when people move out. It can be measured by what is called net migration. Net migration is the difference between the number of people moving into a state and the number of people moving out of the state. If the number is positive, then more people are moving in than out. If the number is negative, then more people are moving out than in.

As you complete the task, keep this question in mind:

How can geometric areas model probability?

For this task, you will apply what you have learned in the lesson to answer this question.

Directions

Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated.

Your teacher will give you further directions about how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or print it and hand in a hard copy.

Now, let’s get started!

Explain why you chose the data sources that you did for this assignment.

Why did you choose your data source about migration and population change per state? (5 points)

Why did you choose your data source about the number of births per state? (5 points)

Analyze the data you gathered about population change in the United States in 2011. Use the following map of the United States as a reference.

Observe the sizes of each state in the cartogram you created. Some are larger and others are smaller. What is indicated by the size of each state? (5 points)

What do you observe about Michigan? What can you conclude about the population change in Michigan during 2011? (5 points)

What can you conclude about the population of Florida? Justify your answer by using the cartogram. (5 points)

Use the data you looked up in the interactive data table regarding the population change in Florida and Michigan during 2011. Do those values match with your conclusions above? (5 points)

The population of the United States increased by 27,323,632 people in 2011. What proportion of this change is accounted for by Florida’s population? Round to the nearest hundredth. (5 points)

Analyze the data you gathered about the number of births in the United States in 2011.

What do you observe about Michigan and Florida? Justify your conclusion by using the cartogram. (5points)

There were 4,008,000 births in the United States during 2011. What proportion of all national births are accounted for by Michigan? What proportion of all national births are accounted for by Florida? (5 points)

Suppose a person born during 2011 in the United States is chosen randomly. Is the person more likely to have been born in Florida or Michigan? Explain. (5 points)

The population of the United States was 308,745,538 in 2011. There were 18,801,310 people in Florida and 9,883,640 people in Michigan. Compute the probability that a randomly selected US resident in 2011 was living in Florida. Do the same for Michigan. Which is greater? (5 points)

The number of births in a region depends mostly on population size. There are also random fluctuations and regional differences. Consider your answers about populations and birth rates. Are your answers consistent with this fact? (5 points)

Analyze the data you gathered about the net migration rate in each state during 2011.

What does it mean if net migration were negative? (5 points)

What can you conclude about net migration in Florida and Michigan during 2011? Use your observations above to justify your answer. (5 points)