9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

4:44 minutes

Problem 13.2.10

Textbook Question

In Exercises 9–12, use the sign test for the claim involving nominal data.
Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use a 0.01 significance level to test the claim that there is a difference between the rate of medical malpractice lawsuits that go to trial and the rate of such lawsuits that are dropped or dismissed.

Verified step by step guidance

Step 1: Understand the problem and identify the hypothesis. The claim is that there is a difference between the rate of medical malpractice lawsuits that go to trial and the rate of those that are dropped or dismissed. This is a two-tailed test because we are testing for a difference, not a specific direction.

Step 2: Define the null and alternative hypotheses. The null hypothesis (H₀) is that the proportion of lawsuits dropped or dismissed is equal to the proportion of lawsuits that go to trial. The alternative hypothesis (H₁) is that the proportions are not equal. In mathematical terms: H₀: p = 0.5 and H₁: p ≠ 0.5.

Step 3: Calculate the test statistic. The sign test is based on the number of positive and negative differences. Here, the number of lawsuits dropped or dismissed is 856, and the total number of lawsuits is 1228. The number of lawsuits that go to trial is 1228 - 856 = 372. Use the smaller of these two values (372) as the test statistic.

Step 4: Determine the critical value. Since the significance level is 0.01 and this is a two-tailed test, divide the significance level by 2 (0.01 / 2 = 0.005) to find the critical region in each tail. Use a binomial distribution table or normal approximation (if applicable) to find the critical value for the given sample size and significance level.

Step 5: Compare the test statistic to the critical value and make a decision. If the test statistic falls within the critical region, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Conclude whether there is sufficient evidence to support the claim that there is a difference between the rates of lawsuits that go to trial and those that are dropped or dismissed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sign Test

The sign test is a non-parametric statistical method used to evaluate the median of a single sample or to compare two related samples. It is particularly useful for nominal data, where the data can be categorized into two groups. The test counts the number of positive and negative signs in the data, allowing researchers to determine if there is a significant difference between the groups without assuming a normal distribution.

Significance Level

The significance level, often denoted as alpha (α), is the threshold used to determine whether to reject the null hypothesis in hypothesis testing. A significance level of 0.01 indicates that there is a 1% risk of concluding that a difference exists when there is none (Type I error). In this context, it means that the results must be very strong to be considered statistically significant.

Nominal Data

Nominal data is a type of categorical data that represents distinct categories without any inherent order or ranking. Examples include gender, race, or the outcome of a lawsuit (dropped or dismissed). In statistical analysis, nominal data is often analyzed using non-parametric tests like the sign test, as traditional parametric tests require interval or ratio data.

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