Question

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. STATE: Is helmet use less common among skiers and snowboarders who have head injuries? Follow the four‑step process to answer the questions about this study. (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.) PLAN: Let p1 and p2 be the proportion of injured skiers and snowboarders who wear helmets and the proportion of uninjured skiers and snowboarders who wear helmets, respectively. Select the correct hypotheses for your test.

Answer 1

Answer:

Step-by-step explanation:

Hello!

To answer the question: Do helmets reduce the risk of head injuries for alpine skiers? two groups were taken into consideration:

Group 1

X₁: Number of skiers that wore a helmet and suffered head injuries out of 578

n₁= 578

Success count: x₁= 96

sample proportion p'₁= x₁/n₁= 96/578= 0.17

Group 2 (Control)

X₂: Number of skiers that wore a helmet and didn't suffer head injuries out of 2992

n₂= 2992

Success count: x₂= 656

Sample proportion p'₂= x₂/n₂= 656/2992= 0.22

Pooled sample proportion $p'= \frac{x_1+x_2}{n_1+n_2}= \frac{96+656}{578+2992}= \frac{752}{3570} = 0.21$

Both variables have a binomial distribution and the parameters of interest are the population proportions of skiers/snowboarders that wore helmets.

To reach a conclusion on whether the use of helmets is less common among skiers/snowboarders that had head injuries, you have to compare both groups, if the claim is true, then the population proportion of the first group should be less than the population proportion of the control group, symbolically: p₁ < p₂

The appropriate statistical test is the two proportion Z-test

The statistic hypotheses are:

H₀: p₁ ≥ p₂

H₁: p₁ < p₂

assuming α: 0.05

$Z= \frac{(p'_1-p'_2)-(p_1-p_2)}{\sqrt{p'[\frac{1}{n_1} +\frac{1}{n_2} ]} }$≈N(0;1)

$Z_{H_0}= \frac{(0.17-0.22)-0}{\sqrt{0.21[\frac{1}{578} +\frac{1}{2992} ]} } = -2.40$

This test is one-tailed to the left, and so is its p-value. You calculate it as the probability of obtaining a value at most as -2.40:

P(Z≤-2.40)= 0.008198

Considering that the p-value is less than the level of significance, the decision is to reject the null hypothesis.

In other words, using a 5% significance level, there is significant evidence to conclude that the population proportion of skiers/snowboarders who got injured and used a helmet is less than the population proportion of skiers/snowboarders that had no injuries and used a helmet.

I hope it helps!