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Chapter 4: Problem 38

A study of the effect of seatbelt use in head-on passenger car collisions found that drivers using a seatbelt had a \(64.1 \%\) survival rate, while drivers not using a seatbelt had a \(41.5 \%\) survival rate. If seatbelts have no effect on survival rate, there is less than a \(0.0001\) chance of getting these results (based on data from "Mortality Reduction with Air Bag and Seat Belt Use in Head-on Passenger Car Collisions," by Crandall, Olson, Sklar, American Journal of Epidemiology, Vol. 153, No. 3 ). What do you conclude?

Short Answer

Expert verified
Using a seatbelt significantly improves survival rates in head-on collisions.

Step by step solution

01

- Understand the Data

Determine the survival rates for both groups: drivers using a seatbelt had a 64.1% survival rate, while drivers not using a seatbelt had a 41.5% survival rate.
02

- Hypothesize

Identify the null hypothesis: seatbelts have no effect on survival rate. Here, the null hypothesis (H0) is that using a seatbelt or not has no difference in survival rates.
03

- Alternative Hypothesis

Identify the alternative hypothesis: seatbelts do have an effect on survival rate. The alternative hypothesis (H1) is that using a seatbelt results in a different survival rate compared to not using a seatbelt.
04

- Statistical Significance

Check the statistical significance level given, which is less than 0.0001. This means there is very strong evidence against the null hypothesis.
05

- Compare and Conclude

Since the p-value (< 0.0001) is much lower than the standard significance level (0.05 or even 0.01), we reject the null hypothesis. Thus, seatbelts do significantly affect survival rates in head-on collisions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The Null Hypothesis
In hypothesis testing, the null hypothesis, often represented as \( H_0 \), is a statement suggesting that there is no effect or no difference. In the context of our study, the null hypothesis asserts that seatbelt use has no significant impact on the survival rates in head-on car collisions.

This presumption of no effect sets the stage for the statistical analysis. By default, we consider the null hypothesis to be true until the evidence (data) suggests otherwise. If the results show a significant difference, we may then reject this hypothesis in favor of the alternative.
The Alternative Hypothesis
Contrary to the null hypothesis is the alternative hypothesis, denoted as \( H_1 \). This is the statement indicating that there is an effect or a difference. For our study, the alternative hypothesis posits that seatbelt use does impact survival rates in head-on car collisions.

Hypothesis tests are designed to assess the strength of evidence against the null hypothesis. If we gather sufficient evidence, we reject the null hypothesis and accept the alternative. In our case, rejecting the null hypothesis would mean concluding that seatbelt use indeed affects survival rates.
Understanding the p-value
The p-value is a crucial aspect of hypothesis testing. It represents the probability of observing data that is at least as extreme as the results actually obtained, assuming that the null hypothesis is true. Basically, it tells us how unusual our results are under the null hypothesis.

For instance, in our seatbelt study, the p-value is less than \(0.0001\). This means there is less than a \(0.01 \)% chance that the observed difference in survival rates occurred due to random variation alone, assuming seatbelts have no effect. Lower p-values indicate stronger evidence against the null hypothesis.
Statistical Significance
Statistical significance is a measure of whether the results of a study are likely due to chance. In hypothesis testing, we compare the p-value to a predetermined significance level (alpha) to decide if the results are statistically significant.

Common alpha levels are 0.05 or 0.01. If the p-value is less than alpha, we reject the null hypothesis. In our study, the p-value is less than \(0.0001\), which is much smaller than typical alpha levels. This strong evidence leads us to conclude that the difference in survival rates is statistically significant, meaning the effect is likely real and not due to chance.
Survival Rate
Survival rate is a key metric in studies evaluating safety interventions, like the use of seatbelts in car collisions. It refers to the proportion of individuals who survive after a specified event.

In this study, drivers using seatbelts had a survival rate of \(64.1\text{\textbackslash}%\), while those not using seatbelts had a survival rate of \(41.5\text{\textbackslash}%\). These rates help quantify the effectiveness of seatbelts in reducing fatalities in head-on collisions.
Higher survival rates among seatbelt users compared to non-users suggest a beneficial effect, leading us to reject the null hypothesis in favor of the alternative.

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