An extensive study of the cost of health care in the United...

An extensive study of the cost of health care in the United States presented data showing that the mean spending per Medicare enrollee in 2003 was $$\$ 6883$$ (Money, Fall 2003). To investigate differences across the country, a researcher took a sample of 40 Medicare enrollees in Indianapolis. For the Indianapolis sample, the mean 2003 Medicare spending was $$\$ 5980$$ and the standard deviation was $$\$ 2518$$. a. State the hypotheses that should be used if we would like to determine whether the mean annual Medicare spending in Indianapolis is lower than the national mean. b. Use the preceding sample results to compute the test statistic and the $p$-value. c. Use $a=.05$. What is your conclusion? d. Repeat the hypothesis test using the critical value approach.

An extensive study of the cost of health care in the United States presented data showing that the mean spending per Medicare enrollee in 2003 was $$\$ 6883$$ (Money, Fall 2003). To investigate differences across the country, a researcher took a sample of 40 Medicare enrollees in Indianapolis. For the Indianapolis sample, the mean 2003 Medicare spending was $$\$ 5980$$ and the standard deviation was $$\$ 2518$$.
a. State the hypotheses that should be used if we would like to determine whether the mean annual Medicare spending in Indianapolis is lower than the national mean.
b. Use the preceding sample results to compute the test statistic and the $p$-value.
c. Use $a=.05$. What is your conclusion?
d. Repeat the hypothesis test using the critical value approach.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical framework used to decide if there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis, which represents no effect or a baseline status quo, and an alternative hypothesis, which represents a deviation from the status quo. Decisions are made based on the outcome of a test statistic and its corresponding p-value compared to a pre-determined significance level.

One-Sample t-Test

The one-sample t-test is a procedure used to compare the mean of a single sample to a known or hypothesized population mean when the population standard deviation is unknown. It is particularly useful when working with small sample sizes. The test statistic is computed using the difference between the sample mean and the hypothesized mean, divided by the standard error of the sample mean, and it follows a t-distribution with degrees of freedom equal to the sample size minus one.

Test Statistic Calculation

The test statistic quantifies how far the sample mean deviates from the hypothesized population mean in terms of the standard error. For a one-sample t-test, the formula is t = (sample mean - hypothesized mean) / (sample standard deviation / ?n), where n is the sample size. This computed value is used to assess the likelihood of observing such a result under the null hypothesis.

P-Value and Significance Level

The p-value indicates the probability of observing a test statistic as extreme as, or more extreme than, what was actually observed, assuming the null hypothesis is true. It helps determine whether the sample provides sufficient evidence to reject the null hypothesis. The significance level (?), often set at 0.05, is the threshold at which the null hypothesis is rejected if the p-value is less than ?.

Critical Value Approach

The critical value approach involves comparing the calculated test statistic to a critical value from the relevant distribution (e.g., t-distribution for a one-sample t-test) corresponding to the chosen significance level. If the test statistic falls in the rejection region (beyond the critical value), the null hypothesis is rejected. This method provides an alternative to using the p-value for making decisions in hypothesis testing.

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