Grades 912
Discover How Two Way Tables Are Used To Determine Auto Insurance Deductibles and Premiums
Standards
Concepts
Students collect data from a handson activity involving Colors Goldfish® Baked Snack Crackers to learn how to create a twoway 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 twoway 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

How Two Way Tables Are Used To Determine Auto Insurance Deductibles And Premiums. PowerPoint presentation  PDF presentation

Large bag of Colors Goldfish® Baked Snack Crackers (enough for each group to get approximately a cup of Goldfish)

Paper plates or paper towels, one for each group

Activity 1 and Activity 2, one for each student

Activity 1 Answer Key and Activity 2 Answer Key, one for the teacher
 Calculator, one for each student
Assessment Activity
Class of Travel  Survived  Died  Total 
First Class  197  122  
Second Class  94  167  
Third Class  151  476  
Total 

According to the data, how many total passengers were on the Titanic?
 442
 765
 995

[1207]

What is the marginal distribution of Third Class passengers who died?
 [39.43%]
 48.37%
 62.22%

75.92%

Of the passengers who survived what is the conditional distribution of those who were First Class?
 61.76%
 54.37%
 [44.57%]

16.32%

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.]
 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

Discuss the following:
 How do you find a marginal distribution? [Divide the total in each cell by the total for the table and then multiply by 100.]

How do you find a conditional distribution? [Divide each value by the amount of the other related value and multiply by 100.]

What is risk? [The chance of loss or harm.]

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.]

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.]

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.]

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 middleaged drivers, older drivers more than middleaged drivers.]
 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
Assessment

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.

Divide students into pairs or groups of three and distribute a copy of Activities 1 and 2 to each student.

Distribute a paper plate or paper towel as well as a cup of “Colors Goldfish® Baked Snack Crackers” to each group.

Display Slide 1 and explain that students will collect data that will allow them to create a twoway table. Explain that a twoway table is a tool used to organize data from two categorical variables.

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.]

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.]

Tell students to count the number of Goldfish in each group.

Show Slide 3 and tell students to enter their totals in their twoway 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.

Give students adequate time to fill in their twoway tables on Activity 1 with their group's data.

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.

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.

Demonstrate how to find marginal distributions for the table on Slide 5.

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.]

Give students adequate time to find the marginal distributions for their group's Goldfish data.

Show Slide 7. Define conditional distribution as the probability that a randomly selected person or item from a subpopulation 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.

Display Slide 8, and demonstrate how to compute conditional distributions for the table.

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.]

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.

Give the students adequate time to calculate totals, marginal and conditional distributions before showing the answers on Slide 10.

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 twoway 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.]

Ask students if they know of any real life examples that use marginal and conditional distributions. [Answers will vary.]

Tell students that insurance companies use information like the data from Activity 2 to determine who is a higher risk when providing insurance.

Tell students that the Center for Disease Control, CDC, reports young people ages 1524 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. (https://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.

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.

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.

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.

Direct students to read the article from Forbes magazine at https://www.forbes.com/sites/moneywisewomen/2013/01/08/whatreallygoesintodeterminingyourinsurancerates/#5ecb64aa3f85 Note: this can be done for homework.

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 middleaged drivers, and older drivers more than middleaged drivers.]

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.
 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.]
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