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Chapter 13: Problem 51

In a study of malpractice claims where a settlement had been reached, two random samples were selected: a random sample of 515 closed malpractice claims that were found not to involve medical errors and a random sample of 889 claims that were found to involve errors (New England Journal of Medicine [2006]: \(2024-2033\) ). The following statement appeared in the paper: "When claims not involving errors were compensated, payments were significantly lower on average than were payments for claims involving errors \((\$ 313,205\) vs. \(\$ 521,560, P=0.004)\) a. What hypotheses did the researchers test to reach the stated conclusion? b. Which of the following could have been the value of the test statistic for the hypothesis test? Explain your reasoning. i. \(\quad t=5.00\) iii. \(t=2.33\) ii. \(\quad t=2.65\) iv. \(l=1.47\)

Short Answer

Expert verified
The hypotheses tested by the researchers are: H鈧�: 渭鈧� = 渭鈧� (no significant difference between average payments) H鈧�: 渭鈧� < 渭鈧� (average payment for claims involving errors is greater than those not involving errors) The possible test statistics could be i. \(t = 5.00\), ii. \(t = 2.33\), or iii. \(t = 2.65\), as these are positive values. Without information on the decision rule based on the sample sizes and significance level, we cannot determine the exact test statistic used in the hypothesis test.

Step by step solution

01

1. Define Null and Alternative Hypotheses

Let 渭鈧� represent the average payment for claims not involving errors, and 渭鈧� represent the average payment for claims involving errors. The null hypothesis (H鈧�) is that there is no significant difference between the average payments for claims not involving errors and those involving errors. The alternative hypothesis (H鈧�) is that there is a significant difference, meaning the average payment for claims involving errors is greater than those not involving errors. H鈧�: 渭鈧� = 渭鈧� (no significant difference between average payments) H鈧�: 渭鈧� < 渭鈧� (average payment for claims involving errors is greater than those not involving errors)
02

2. Determine the Test Statistic

The researchers found the p-value to be 0.004, which is less than the typical significance level of 0.05. This indicates that there is strong evidence against the null hypothesis, so we can reject it. In this case, we expect a positive test statistic, as the average payment for claims involving errors is greater compared to those not involving errors. Now, let's evaluate the given test statistics: i. \(t = 5.00\): This is a positive test statistic and, if it is large enough relative to the sample sizes, could lead us to reject the null hypothesis in favor of the alternative hypothesis. ii. \(t = 2.33\): Like \(t = 5.00\), this is also a positive test statistic and may be large enough to lead us to reject the null hypothesis. iii. \(t = 2.65\): Again, this is also a positive test statistic and may be large enough to lead us to reject the null hypothesis. iv. \(l = 1.47\): This value is not correct, as it should be represented as "t" instead of "l." Out of the given options, the test statistic could possibly be i. \(t = 5.00\), ii. \(t = 2.33\), or iii. \(t = 2.65\), as these are positive values. To determine which of these is the correct test statistic, one would need to know the decision rule based on the sample sizes and significance level. However, based on the information given in the exercise, any of these three test statistics could have been the value used in the hypothesis test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
The null hypothesis is a fundamental concept in hypothesis testing. It is a statement that suggests no significant change, effect, or difference exists in the context being analyzed. In hypothesis testing, the purpose is to test the validity of this hypothesis by collecting data and analyzing it.

In the exercise you encountered, the null hypothesis (denoted as \(H_0\)) asserts that there is no difference in average payments for malpractice claims, whether they involve errors or not. Mathematically, this is expressed as \( \mu_1 = \mu_2 \), indicating that the average payment for claims without errors (\( \mu_1 \)) is equal to the average payment for claims with errors (\( \mu_2 \)).

When performing hypothesis testing, researchers start with the assumption that the null hypothesis is true. Data is then collected and analyzed to determine if there is sufficient evidence to reject this assumption. If the evidence is strong, derived from data like the test statistic or p-value, the null hypothesis might be rejected.
Alternative Hypothesis
The alternative hypothesis (denoted as \(H_1\)) is the counterpart to the null hypothesis in hypothesis testing. It represents the statement that researchers aim to support, suggesting a significant effect or difference exists.

In this specific exercise, the alternative hypothesis posits that there is indeed a significant difference in the average payments between malpractice claims not involving errors and those involving errors. Specifically, it claims that the average payment for error-involving claims is greater than for non-error ones, mathematically expressed as \( \mu_1 < \mu_2 \).

The role of the alternative hypothesis is crucial during analysis. If the statistical evidence gathered (e.g., p-value, test statistic) is significant enough, it can lead researchers to reject the null hypothesis and conclude that the alternative hypothesis is supported. This is exactly what happened in the exercise when the researchers concluded that payments were significantly lower for claims not involving errors compared to those with errors.
Test Statistic
The test statistic is a crucial concept in hypothesis testing that helps researchers determine whether to reject the null hypothesis. It is a standardized value calculated from sample data during a hypothesis test. This statistic allows for comparison against a theoretical distribution to infer results.

In hypothesis testing, the choice of test statistic depends on the data and the type of analysis required. Common examples include the "t" statistic in t-tests or the "z" statistic in z-tests. The exercise provided potential candidates for the test statistic values as "t = 5.00", "t = 2.33", and "t = 2.65," all indicating positive differences.

When calculating a test statistic, researchers compare it against critical values determined by significance levels (often 0.05 or 0.01). For instance, if the p-value associated with a test statistic is less than the significance level, the null hypothesis is rejected.
  • A large positive test statistic typically indicates strong evidence against the null hypothesis.
  • In the exercise, the final choice of test statistic among "t = 5.00", "t = 2.33", or "t = 2.65," depends on additional factors like sample sizes and the specific hypothesis test used.
Through the test statistic, researchers could confirm that the average payment for claims involving errors was significantly different from that for claims not involving errors.

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