The research department at the home office of New Hampshire...

The research department at the home office of New Hampshire Insurance conducts ongoing research on the causes of automobile accidents, the characteristics of the drivers, and so on. A random sample of 400 policies written on single persons revealed 120 had at least one accident in the previous three-year period. Similarly, a sample of 600 policies written on married persons revealed that 150 had been in at least one accident. At the . 05 significance level, is there a significant difference in the proportions of single and married persons having an accident during a three-year period?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical framework used to make decisions about a population based on sample data. It involves formulating a null hypothesis that represents a default position (usually no effect or no difference) and an alternative hypothesis that indicates the presence of an effect or difference. Statistical evidence from sample data is used to assess whether the observed result is unlikely under the assumption that the null hypothesis is true.

Two-Proportion Z-Test

A two-proportion z-test is a specific form of hypothesis testing that compares the proportions of a binary outcome in two independent groups. This test determines if there is a statistically significant difference between the two observed proportions by standardizing the difference using an estimate of the variability, or standard error, under the assumption that the null hypothesis (equal proportions) is true.

Standard Error and Pooled Proportion

The standard error in the context of comparing two proportions quantifies the variability expected in the difference between sample proportions due to random sampling. When assessing the null hypothesis that the two population proportions are equal, a pooled proportion is calculated as a combined estimate from both samples. This pooled estimate is then used to compute the standard error used in the z-test.

Significance Level and Critical Values

The significance level, often set at a value like 0.05, defines the threshold for rejecting the null hypothesis by controlling the probability of committing a Type I error, which is the mistake of rejecting a true null hypothesis. Critical values, derived from a statistical distribution such as the standard normal distribution, serve as boundaries; if the test statistic falls beyond these boundaries, the null hypothesis is rejected in favor of the alternative hypothesis.

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