The president of the American Insurance Institute wants to...

The president of the American Insurance Institute wants to compare the yearly costs of auto insurance offered by two leading companies. He selects a sample of 15 families, some with only a single insured driver, others with several teenage drivers, and pays each family a stipend to contact the two companies and ask for a price quote. To make the data comparable, certain features, such as the amount deductible and limits of liability, are standardized. The sample information is reported below. At the .10 significance level, can we conclude that there is a difference in the amounts quoted?

Key Concepts

Test Statistic and p-value

The test statistic, such as the t-statistic in a paired t-test, measures the magnitude of the observed difference relative to the variability in the differences. Its corresponding p-value indicates the probability of observing the data (or something more extreme) if the null hypothesis were true, which is used to decide whether to reject the null hypothesis.

Significance Level

The significance level, denoted as alpha, is the threshold used for deciding whether to reject the null hypothesis. In this example, an alpha level of 0.10 indicates that there is a 10% risk of rejecting the null hypothesis when it is actually true, and it sets the criterion for determining statistical significance.

Data Standardization

Data standardization involves controlling or adjusting certain factors to make comparisons fair and valid across different measurements. In the given scenario, standardizing features such as deductible amounts and liability limits ensures that any observed differences in quotes are attributable to the companies' pricing rather than extraneous variables.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter by comparing an observed sample statistic against a null hypothesis. In this context, the test is used to determine whether there is a significant difference in the insurance quotes from the two companies at a pre-specified significance level by evaluating p-values and test statistics.

Paired Sample Analysis

Paired sample analysis is used when data observations are naturally paired or matched, such as when the same subjects provide two different measurements. In this question, because the same set of families provided quotes from both insurance companies, the paired design controls for potential confounding factors by comparing the differences within the same family.

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