Are Seat Belts Effective? A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed (based on data from “Who Wants Airbags?” by Meyer and Finney, Chance, Vol. 18, No. 2). We want to use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities.


a. Test the claim using a hypothesis test.

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that the fatality rates for occupants wearing seat belts and not wearing seat belts are the same. Mathematically, H₀: p₁ = p₂, where p₁ is the fatality rate for occupants not wearing seat belts, and p₂ is the fatality rate for occupants wearing seat belts. The alternative hypothesis (H₁) states that the fatality rate for occupants not wearing seat belts is higher than for those wearing seat belts. Mathematically, H₁: p₁ > p₂.
2. Step 2: Identify the sample proportions and sample sizes. For occupants not wearing seat belts, the sample size is n₁ = 2823, and the number of fatalities is x₁ = 31. The sample proportion is calculated as p̂₁ = x₁ / n₁. For occupants wearing seat belts, the sample size is n₂ = 7765, and the number of fatalities is x₂ = 16. The sample proportion is calculated as p̂₂ = x₂ / n₂.
3. Step 3: Calculate the pooled proportion (p̂) under the assumption that the null hypothesis is true. The pooled proportion is given by the formula: p̂ = (x₁ + x₂) / (n₁ + n₂). This represents the overall fatality rate across both groups combined.
4. Step 4: Compute the test statistic using the formula for the z-test for two proportions: z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]. Substitute the values of p̂₁, p̂₂, p̂, n₁, and n₂ into the formula to calculate the z-value.
5. Step 5: Determine the critical value and compare it to the test statistic. For a one-tailed test at a significance level of α = 0.05, find the critical z-value from the standard normal distribution table. If the calculated z-value exceeds the critical z-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Conclude whether the data supports the claim that seat belts are effective in reducing fatalities.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or no difference, and an alternative hypothesis (H1) that indicates the presence of an effect. The test assesses the likelihood of observing the sample data if the null hypothesis is true, leading to a decision to either reject or fail to reject H0 based on a predetermined significance level.

Significance Level

The significance level, denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis in a hypothesis test. Commonly set at 0.05, it represents a 5% risk of concluding that a difference exists when there is none (Type I error). In this context, using a 0.05 significance level means that if the probability of observing the data under the null hypothesis is less than 5%, the null hypothesis will be rejected, suggesting that seat belts are effective.

P-Value

The p-value is a statistical measure that helps determine the significance of the results from a hypothesis test. It quantifies the probability of obtaining results at least as extreme as the observed data, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis. In this scenario, comparing the p-value to the significance level will help decide whether the data supports the claim that seat belts reduce fatalities.