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# Exponential Distributions

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Practice Exponential Distributions
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# Exponential Distribution - Answer Key

## Battery Life and Exponential Distributions

### Topic

Battery Life and Exponential Distributions

### Vocabulary

• Exponential Distribution
• Continuous Data
• Coefficient of Determination

### Student Exploration

#### Sick of replacing the batteries in your household objects? How often do you have to replace the batteries in your video game controllers? In your camera?

People use AA and AAA batteries for all types of household objects. Batteries power video game controllers, cameras, toys, calculators, portable radios, flashlights, and small appliances. Unfortunately batteries don’t last very long. The battery sales across the globe make about \$50 billion (U.S.) each year, and is a growing industry, see http://batteryuniversity.com/learn/article/battery_statistics. The life of a battery can be represented as an exponential distribution, when measuring the probability of a battery dying over time. Check out the following table tracking the days of use of batteries and the probability of failure over that time, note that the probability of failure is in decimal form, such that 0.095 means 9.5%.

Table of the probability of the failure of a battery over time (days of use).
Days in Use Probability of Failure
1 0.00995
2 0.0198
5 0.04877
10 0.095
18 0.16473
32 0.27385
43 0.34295
99 0.62842

1. Use your TI-84 calculator to create an exponential regression equation to represent the exponential distribution. The exponential regression equation is $y=0.0388894342(1.036044635)^x$.
2. What is the coefficient of determination? The coefficient of determination (also referred to as the correlation coefficient) is .652900134.
3. How likely is it that the exponential equation is correct? The exponential regression equation is not a very accurate as the coefficient of determination is fairly far from 1. You can also see this when graphing the equation $y=0.0388894342(1.036044635)^x$ on the same $x-$ and $y-$axis as the original table of values.
4. What is the probability that a battery will fail after 150 days? The probability that the battery will fail after 150 days is 78.81 because $0.0388894342(1.036044635)^{150} = 78.81$.
5. What are the implications of this information when thinking of your uses of batteries?

### Extension Investigation

Other applications of exponential distributions include representing the traffic patterns in a city, the length of support phone calls, life of many electrical items, or the number of customer transactions at a store. Auto insurers use exponential distributions to determine the risk of a client getting in an accident. This is how they determine how much to charge individuals for insurance. Check out this practice problems for the test for people who want to become risk assessors for insurance companies, “Example P Practice Problems” 7, 6, 4, 3, 2, and 1 at http://probabilityexam.wordpress.com/tag/exponential-distribution/ .

### Connections to other CK-12 Subject Areas

• Binomial Distributions and Probability
• Multinomial Distribution
• Linear Regression Equations