# Do Seat Belts Save Lives?

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## Abstract

The purpose of this observational study is to determine if wearing seat belts help save lives in car accidents based on government records of seat belt usage and corresponding fatality rates in car accidents.

## Introduction

The concept of using a seat belt in a moving vehicle was first thought of in 1849. However, it did not become popular until the three-point seat belt was patented by the Swedish inventor Nils Bohlin and introduced by Volvo in 1959. This model, used in the modern day, consists of three attachment points, the shoulder and both hips. Its job is to protect an occupant from injury in the event of a car accident. In terms of physics, a seat belt increases the stopping distance of an occupant during a car crash. According to the work-energy principle, this lessens the impact force applied to the occupant.

Work = Δ Kinetic Energy

(Force)(distance) = Δ½(mass)(velocity)^{2}

Since the change in K remains the same, an increase in distance decreases the force acted on the occupant of the vehicle. This principle is believed to save lives in car accidents.

The physics of this concept is rather simple, but it does not account for other factors that may be involved in a car accident. Therefore, using statistical analysis of government data on car crashes and their corresponding fatality rate, this study will test the idea if seat belts actually save lives in car accidents.

## Procedure

- Collect data on car accidents and corresponding fatality rates from the CDS (Crashworthiness Data System) in the US government.
- Separate data in to two groups, those who wore seat belts and those who did not wear seat belts.
- Check appropriate conditions/assumptions and perform a hypothesis test, specifically a two-proportion z-test (difference in proportions), on the data.

## Analysis

The data and percentages complied can be seen in the following table. The following are the procedures carried out for a two-proportion z-test.

### Hypothesis:

Where

H_{0}: p_{1}–p_{2} = 0

H_{a}: p_{1}–p_{2} > 0

p_{1} = percentage who died out of sample of car crashes where seat belt was not worn.

p_{2} = percentage who died out of sample of car crashes where seat belt was worn.

### Conditions/Assumptions:

*Independence*: car crashes were divided into two samples, those who wore seat belts and those who didn't. Each car crash occurred at different times so they could not affect each other.*Simple Random Sample*: the two samples were randomly chosen from data of all car crashes in the US compiled by the government.*10% condition*: (10)(total sample) < total population. It is assumed that (10)(3378) is less than all of the car crashes in the US.*Success/Failure Condition*:

n_{1} p_{1} > 10 → 657 > 10 |
n_{2} p_{2} > 10 → 1287 > 10 |

n_{1} q_{1} > 10 → 657 > 10 |
n_{2} q_{2} > 10 → 770 > 10 |

### 2 Proportion Z Test:

Where

z score = (p_{1}–p_{2})/[(pq)(1/n_{1} + 1/ n_{2})]

p = (x_{1} + x_{2})/(n_{1} + n_{2}) = (657 + 1287)/(1321 + 2057) = 1944/3378 = 0.575

z score = (1287/2057–657/1321)/[(0.575)(1/0.575)(1/2057 + 1/1321)]

z score = 7.36

A z score represents the probability that H_{0} is true in terms of the number of standard deviations from the center of a normal curve (7.36 is extremely far away).

Using the calculated z- score, a two-proportion z test yields a p–value = 9.05 × 10^{−14}.

- The p-value gives the probability that H
_{0}is true.

## Conclusion

Since the p-value is equal to 9.05 × 10^{−14} is much less than 0.05 (a reasonable alpha), there is sufficient evidence to reject H_{0}. Therefore, I accept the alternative hypothesis that the percentage that died out of sample of car crashes where seat belt was not worn is greater than the percentage that died out of sample of car crashes where seat belt was worn. Furthermore, these results give evidence that seat belts do save lives in car crashes.

## Sources of Error

- As seen on the data table compiled, one sample size is twice as large as the other. This may have a confounding yet marginal effect on the results since all of the conditions and assumptions were satisfied for this hypothesis test.
- Factors such as gender, age, and airbag deployment were not accounted for in this study. They may of had an effect on the results, but for the purpose of this study I wanted to look at the specific effect of seat belt usage and its direct relationship on fatality rates in car accidents.

Evan Kaplan -- 2005

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