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Chapter 23: Problem 5

{ Protecting Skiers and Snowboarders. Most alpine }\( skiers and snowboarders do not use helmets. Do helmets reduce the risk of head injuries? A study in Norway compared skiers and snowboarders who suffered head injuries with a control group who were not injured. Of 578 injured subjects, 96 had worn a helmet. Of the 2992 in the control group, 656 wore helmets. 9 Is helmet use less common among skiers and snowboarders who have head injuries? Follow the four-step process as illustrated in Examples 23.4 and \)23.5$ (pages 524 and 525 ). (Note that this is an observational study that compares injured and uninjured subjects. An experiment that assigned subjects to helmet and no- helmet groups would be more convincing.)

Short Answer

Expert verified
Helmet use is less common among injured skiers and snowboarders.

Step by step solution

01

State Hypotheses

We want to compare helmet usage between injured and uninjured skiers/snowboarders. Our null hypothesis, \( H_0 \), is that helmet use is the same for injured and uninjured subjects. Our alternative hypothesis, \( H_a \), is that helmet use is less common among those with head injuries. Mathematically, \( H_0: p_1 = p_2 \) and \( H_a: p_1 < p_2 \), where \( p_1 \) is the proportion of injured subjects who wore helmets, and \( p_2 \) is the proportion of uninjured subjects who wore helmets.
02

Calculate Proportions

Calculate the proportion of injured subjects who wore helmets: \( p_1 = \frac{96}{578} \approx 0.166 \). For the control group: \( p_2 = \frac{656}{2992} \approx 0.219 \). These proportions suggest that helmet use is less common among injured subjects.
03

Determine Test Statistic

We will use a two-proportion z-test. Calculate the pooled sample proportion \( \hat{p} = \frac{96 + 656}{578 + 2992} = \frac{752}{3570} \approx 0.210 \). Now, calculate the z-statistic: \[ z = \frac{p_1 - p_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{578} + \frac{1}{2992}\right)}} \]. Plug in the numbers to find the z-value.
04

Make a Decision

Based on the z-value calculated, compare it to the standard normal distribution (using a z-table or calculator) to find the p-value. If the p-value is less than the significance level (commonly 0.05), we reject the null hypothesis, indicating helmet use is indeed less common among those with head injuries.

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Key Concepts

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

Two-Proportion Z-Test
The Two-Proportion Z-Test is a statistical method used when you want to compare two proportions to see if there is a significant difference between them. It's perfect for cases where you have two groups and you are interested in understanding whether there is a difference in a particular trait or behavior, such as helmet usage among different groups of skiers and snowboarders in our study.

In the skiers and snowboarders' scenario, we have two groups:
  • Injured skiers/snowboarders
  • Uninjured skiers/snowboarders
We calculated the proportion of helmet use in both groups.
  • For injured subjects, 96 out of 578 wore a helmet.
  • For the control (uninjured) group, 656 out of 2992 wore helmets.
Once we know these proportions, we use the Two-Proportion Z-Test to determine if the observed difference in helmet use between these two groups is statistically significant, meaning it's unlikely to be due to random chance. By calculating the z-statistic and comparing it to the standard normal distribution, we can conclude if there's evidence to suggest a real difference in behaviors or characteristics between these two groups.
Observational Study
In an observational study, researchers observe and collect data about events as they naturally occur without manipulating any variables. This type of study is essential in cases where it's unethical or impractical to conduct a controlled experiment. For example, in our scenario, it would not be ethical to make some skiers or snowboarders wear helmets and others go without, especially in potentially dangerous situations.

An observational study can be great for identifying associations, like understanding if there's a link between helmet use and head injuries among skiers and snowboarders. It aims to gather data on existing conditions, allowing researchers to infer whether there is a correlation.
  • However, remember that correlation does not imply causation.
  • Variables that are not controlled can affect the results, such as skill level or the skiing environment.
  • A carefully designed observational study can still provide valuable insights and help shape further research and experiments.
Null and Alternative Hypotheses
Formulating the null and alternative hypotheses is a crucial first step in any hypothesis testing process. These hypotheses provide a framework for testing whether observed data reflect actual differences or merely randomness.

For our study:
  • The **null hypothesis** (\( H_0 \)) suggests that there is no difference in helmet usage between injured and uninjured skiers/snowboarders, implying any observed difference is due to sampling variability.
  • The **alternative hypothesis** (\( H_a \)) indicates we believe there is a difference 鈥� specifically, that helmet use is less common among those with head injuries.
By testing these hypotheses, researchers can systematically evaluate the likelihood that the observed data occurred under the null hypothesis. If data suggest that helmet use is significantly less common among the injured, the null hypothesis can be rejected, leading to the conclusion that the alternative hypothesis may be true. This decision is usually based on a p-value derived from the test statistic and a pre-determined significance level, typically 0.05.

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Most popular questions from this chapter

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