You have been asked to evaluate single-employer plans after...

You have been asked to evaluate single-employer plans after the establishment of the Health Benefit Guarantee Corporation. A random sample of 76 percentage changes in promised health benefits was observed. The sample mean percentage change was 0.078 , and the sample standard deviation was 0.201 . Find and interpret the $p$-value of a test of the null hypothesis that the population mean percentage change is 0 against a two-sided alternative.

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Key Concepts

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. In the context of a two-sided test, it reflects the total probability in both tails of the distribution. A very small p-value indicates that the observed data is unlikely under the null hypothesis, thus lending support to the alternative hypothesis.

t-Statistic and t-Distribution

The t-statistic quantifies the difference between the sample mean and the hypothesized population mean relative to the variability in the sample, scaled by the standard error. When the population standard deviation is unknown, the t-distribution, which adjusts for sample size through degrees of freedom, is used to determine the likelihood of observing a test statistic as extreme or more extreme than the calculated value.

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to evaluate whether observed data sufficiently contradicts a stated null hypothesis. This involves comparing the calculated test statistic against a theoretical probability distribution, which determines whether the null hypothesis can be rejected in favor of an alternative hypothesis.

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