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Discover How Two Way Tables Are Used To Determine Auto Insurance Deductibles and Premiums

Students collect data from a hands-on activity involving Colors Goldfish® Baked Snack Crackers to learn how to create a two-way table, find the marginal distribution, and find the conditional distribution. Students then use given data to find marginal and conditional distributions to determine which groups are at higher risks of receiving a traffic ticket. Lastly, students use this information and investigate other factors to determine how insurance companies compute auto insurance deductibles and premium payments.

Time Required

60 minutes

Will Be Able To

  • Create a two-way table.
  • Find marginal distributions. 
  • Find conditional distributions.
  • Analyze data to determine high, medium and low risk groups. 
  • Define insurance, premiums, and deductible.
  • Explain why different people pay different amounts in insurance premiums and deductibles.

Materials

Assessment Activity

Use the data below about the 1912 Titanic passengers to answer questions 1-3.
Class of Travel Survived Died Total
First Class 197 122  
Second Class 94 167  
Third Class 151 476  
Total      
 
  1. According to the data, how many total passengers were on the Titanic?

    1. 442
    2. 765
    3. 995
    4. [1207]
       
  2. What is the marginal distribution of Third Class passengers who died?

    1. [39.43%]
    2. 48.37%
    3. 62.22%
    4. 75.92%
       
  3. Of the passengers who survived what is the conditional distribution of those who were First Class?

    1. 61.76%
    2. 54.37%
    3. [44.57%]
    4. 16.32%
       
  4. Why do you think an insurance company would ask a potential customer how many speeding tickets they received in the past five years? [Being given a speeding ticket is a sign that a driver is a higher risk for auto insurance companies. If the potential customer is higher risk, his or her premiums and deductibles would be higher.
     
  5. Why do parents often require their own teenage children to attend a driver’s safety or driver’s education class? [Parents often pay for their teenage children’s insurance premiums. By attending a driver’s safety or driver’s education class, insurance companies view the driver as a lower risk and consequently charge lower amounts for premiums.]

Conclusion

  1. Discuss the following:

    1. How do you find a marginal distribution? [Divide the total in each cell by the total for the table and then multiply by 100.]
    2. How do you find a conditional distribution? [Divide each value by the amount of the other related value and multiply by 100.]
       
  2. What is risk? [The chance of loss or harm.]
     
  3. What is insurance? [An agreement between an insurance company and a policyholder that allows the policyholder to transfer risk by paying a fee now to avoid the possibility of a larger loss later.]
     
  4. How is risk related to premiums and deductibles that people pay for insurance? [Level of risk and amounts charged for premiums and deductibles by insurance companies are directly correlated. The higher the risk, the higher premiums and deductibles will be. People can sometimes reduce premiums by agreeing to pay higher deductibles.]
     
  5. What is automobile insurance? [Auto insurance is a type of property insurance that pays for damages or losses to the insured’s car. Auto insurance often includes liability coverage for actions of the insured that cause harm to other people or their property.]
     
  6. Which groups present higher risk when it comes to auto insurance? [Men present greater risk than women, teenage males more risk than teenage girls, younger drivers more than middle-aged drivers, older drivers more than middle-aged drivers.]
     
  7. What behaviors can you control in order to be lower risk for automobile insurance? [Control speed, control whether you drink and drive, control whether you text and drive, and choose whether to take driver education classes.]

Overview

"Risk" is the chance of loss or harm. Personal financial risk exists when unexpected events can damage health, income, property wealth or future opportunities. Insurance is a product that allows people to pay a fee, called a premium, upfront to transfer the costs of a potential loss to a third party. Insurance companies analyze the outcomes of individuals who face similar types of risks to create insurance contracts (policies). By collecting a relatively small amount of money, called a premium, from each policyholder on a regular basis, the company creates a pool of funds to compensate those individuals who experience a large loss. Insurance companies charge high premiums to cover high-risk individuals and events because the risk of monetary loss is greater for these individuals and events. People can lower insurance premiums by behaving in ways that show they pose a lower risk and by assuming higher deductibles. A deductible is a fixed amount an insured person must pay per loss before the insurance company will pay a claim. Auto insurance is a type of property insurance that pays for damage or loss to the insured’s car and often includes liability coverage for actions of the insured which cause harm to other people or their property. Insurance companies consider a number of factors when determining the level of risk associated with providing car insurance to various groups of people. In this lesson, students examine age, gender and driving record using two-way tables. 
 
A two-way table is a tool used to organize data from two categorical variables. A categorical variable is a variable that can take on one of a limited and usually fixed number of possible values. Two-way tables often summarize large amounts of information comparing two sets of data organized in rows and columns.
 
Once students have the data, marginal and conditional distributions can be computed and used to compare risk levels for those seeking insurance. The marginal distribution of one of the categorical variables in a two-way table is the distribution of values of that variable among all individuals described by the table. A conditional distribution of a variable is the distribution of values of that variable among only individuals who have a given value of the other variable. There is a separate conditional distribution for each value of the other variable (faculty.etsu.edu/gardnerr/1530/chapter6.pdf). A teenage male is a high risk for auto insurance companies (an example of marginal distribution). However, of those male teenager drivers, those who take a driver safety class are a lower risk (an example of conditional distribution). Students will use a two-way table and marginal and conditional distributions to determine which group or groups are a higher risk for auto insurance companies and therefore will be charged higher premiums and deductibles. 
 
Note to teachers: the term marginal in this lesson is used differently from the customary use of the term in economics. In economics, marginal refers to the additional or extra of something, such as the additional or marginal cost of production.

Assessment

  1. Tell students that the purpose of this activity is to use data collected from a bag of Colors Goldfish® Baked Snack Crackers to create a marginal distribution and a conditional distribution. 
     
  2. Divide students into pairs or groups of three and distribute a copy of Activities 1 and 2 to each student. 
     
  3. Distribute a paper plate or paper towel as well as a cup of “Colors Goldfish® Baked Snack Crackers” to each group.
     
  4. Display Slide 1 and explain that students will collect data that will allow them to create a two-way table. Explain that a two-way table is a tool used to organize data from two categorical variables.
     
  5. Show Slide 2 and review the data collection. [Teacher note: it is important to use “Colors Goldfish® Baked Snack Crackers” because goldfish come in four different colors – yellow, orange, red and green.]
     
  6. Tell students that after they have separated the goldfish by color they should divide each color group into two categories—goldfish with faces and goldfish without faces. [Teacher note: there will be a total of eight individual groups. Yellow with a face, Yellow without a face, Orange with a face, Orange without a face, Red with a face, Red without a face, Green with a face, Green without a face.]
     
  7. Tell students to count the number of Goldfish in each group.
     
  8. Show Slide 3 and tell students to enter their totals in their two-way table on Activity 1. Point out that each group’s distribution of colors, faces and no faces will be different so each group’s answers will be different. 
     
  9. Give students adequate time to fill in their two-way tables on Activity 1 with their group's data.
     
  10. Show Slide 4 and tell students to total the number of Goldfish for each color adding the rows. Total the number of Goldfish for face/no face, adding the columns. Point out again that each group’s answers will vary.
     
  11. Direct the students to question 2 on Activity 1. Show Slide 5 and define marginal distribution as the distribution of one of the values of one variable among all individuals described by the table. Instruct students to record the definition on Activity 1.
     
  12. Demonstrate how to find marginal distributions for the table on Slide 5.
     
  13. Show Slide 6 and point out that the slide shows the marginal distribution for the sample data given. [Teacher note: because of rounding, percentages may not add up to exactly 100%. In the sample data shown in Slide 6, the percentages in the Total column and row add up to 99.99%. Note: marginal and conditional distributions can be shown as percentages as well as decimals.]
     
  14. Give students adequate time to find the marginal distributions for their group's Goldfish data. 
     
  15. Show Slide 7. Define conditional distribution as the probability that a randomly selected person or item from a sub-population has the one characteristic of interest. Instruct students to record the definition in number 3 on Activity 1. Go over the example on Slide 7 and explain how to determine the conditional distribution for the example.
     
  16. Display Slide 8, and demonstrate how to compute conditional distributions for the table.
     
  17. Instruct students to calculate conditional distribution for Goldfish “Without a face.” Give adequate time for students to calculate the conditional distributions. [Teacher note: point out to students that there are now two conditions for the categories they are counting: color and face. Rounding percentages may not add up exactly to 100%. The percentages add up to 99.99% in the sample data shown in the PowerPoint.]
     
  18. Show Slide 9 and review the scenario. Direct students to calculate the totals, marginal distribution and conditional distribution using the information in the three tables on Activity 2.
     
  19. Give the students adequate time to calculate totals, marginal and conditional distributions before showing the answers on Slide 10.
     
  20. Ask students what the difference is between marginal distribution and conditional distribution. [Marginal distribution is the distribution of values of a variable among all individuals described by a two-way table. A conditional distribution of a variable is the distribution of values of that variable among only individuals who have a given value of the other variable.]
     
  21. Ask students if they know of any real life examples that use marginal and conditional distributions. [Answers will vary.]
     
  22. Tell students that insurance companies use information like the data from Activity 2 to determine who is a higher risk when providing insurance.
     
  23. Tell students that the Center for Disease Control, CDC, reports young people ages 15-24 represent only 14% of the U.S. population. However, they account for 30% ($19 billion) of the total costs of motor vehicle injuries among males and 28% ($7 billion) of the total costs of motor vehicle injuries among females.  (www.cdc.gov/motorvehiclesafety/teen_drivers/teendrivers_factsheet.html).  

    For that reason, insurance companies use distributions like these to determine if someone is a high risk of getting in an accident and costing insurance companies money. 
     

  24. Show Slide 11. Define risk as the chance of loss or harm. If people are more likely to incur loss or harm, insurance companies are more likely to have to pay claims on those people. Explain that insurance allows people to transfer risk by paying a fee now to avoid the possibility of a larger loss later. The price of insurance is influenced by an individual’s behavior. Insurance companies analyze the outcomes of individuals who face similar types of risks to create insurance policies. 
     
  25. Use Slide 12 and explain that insurance companies collect a relatively small amount of money, called a premium, for each policyholder on a regular basis. This allows the company to create a pool of funds to compensate those individuals who experience loss. Define deductible as a fixed amount an insured person must pay per loss before the insurance company will pay a claim.
     
  26. Explain to students that premiums and deductibles are directly correlated to the level of risk an insured person poses; that is, how likely the person is to incur or cause loss or harm, and thus make a claim with the insurance company. Premiums and deductibles are based on the level of risk a policyholder is judged to be. The higher the risk, the higher premiums and deductibles will be. Often policyholders agree to pay higher deductibles in order to reduce their premiums.
     
  27. Direct students to read the article from Forbes magazine at www.forbes.com/sites/moneywisewomen/2013/01/08/what-really-goes-into-determining-your-insurance-rates/ Note: this can be done for homework.
     
  28. Ask students, based on the principle that higher risk groups pay more for insurance, what groups in the article can be assumed to be higher risks. [Men more than women, teenage males more than teenage females, younger drivers more than middle-aged drivers, and older drivers more than middle-aged drivers.]
     
  29. Display Slide 13. Review with students the variables insurance companies consider when determining an individual’s level of risk.

    • Driving record—The number one factor when determining risk is an individual’s driving record. Insurance companies look at how many accidents and/or driving violations an individual has.
    • Commute—The further one lives from work the more he or she will be driving and the greater possibility of being involved in an accident.
      Type of car—Someone with a sports car is more likely to engage in more risky driving habits.
    • Age—Individuals under 25 and over 64 are more likely to be in an automobile accident.
    • Credit score—Individuals with poor credit scores will likely be charged more for insurance than those with higher credit scores.
    • Martial status—On average, married drivers are at lower risk of accidents than unmarried drivers.
    • Gender—Men are a greater risk for insurance companies because they are statistically more likely to get into accidents than women.
    • Grades—Point out that students with good grades are considered more responsible and less likely to engage in risky driving. They are usually eligible for discounts.
       
  30. Ask students what behaviors they can control in order to be considered lower risk for automobile insurance? [Whether they take a driver education class, whether they drink and drive, whether they text and drive, whether they drive while distracted, whether they get good grades, what type of car they drive, whether they maintain a good credit score.]