91Ó°ÊÓ

A hospital readmission is an episode when a patient who has been discharged from a hospital is readmitted again within a certain period. Nationally the readmission rate for patients with pneumonia is \(17 \% .\) A hospital was interested in knowing whether their readmission rate for pneumonia was less than the national percentage. They found 11 patients out of 70 treated for pneumonia in a two-month period were readmitted. a. What is \(\hat{p}\), the sample proportion of readmission? b. Write the null and alternative hypotheses. c. Find the value of the test statistic and explain it in context. d. The p-value associated with this test statistic is \(0.39\). Explain the meaning of the p-value in this context. Based on this result, does the \(\mathrm{p}\) -value indicate the null hypothesis should be doubted?

Step by step solution

01

Calculate the Sample Proportion

The sample proportion \(\hat{p}\) is the number of positive outcomes (in this case, readmitted patients) divided by the total number of outcomes (in this case, patients treated). Here, 11 patients out of 70 were readmitted, hence \(\hat{p} = \frac{11}{70} = 0.1571\). This is the proportion of readmitted patients in the sample.

02

Formulate the Hypotheses

The null hypothesis (H0) is the statement that we are aiming to find evidence against. The alternative hypothesis (Ha) is a statement that we will accept if our data provide enough evidence to reject the H0. Here, H0: p=0.17 and Ha: p<0.17 where p is the population proportion of readmissions.

03

Calculate the Test Statistic

The test statistic for a hypothesis test about a population proportion is given by the formula: \( Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\), where \(\hat{p}\) is the sample proportion, p is the population proportion under the null hypothesis, and n is the sample size. Plugging \(\hat{p}=0.1571\), p=0.17, and n=70 into the formula gives \( Z = \frac{0.1571-0.17}{\sqrt{\frac{0.17(1-0.17)}{70}}} = -0.8641\). The negative value of the test statistic indicates that the sample proportion is less than the population proportion under the null hypothesis.

04

Interpret the P-Value

A p-value of 0.39 means that if the null hypothesis were true, there would be a 39% chance of obtaining a sample proportion as extreme or more extreme as the one we got. It does not provide strong evidence against the null hypothesis. Since 0.39 > 0.05 (common threshold), we would not reject the null hypothesis. Therefore, the p-value does not support the claim that the hospital's readmission rate is lower than the national rate.

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Key Concepts

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

Sample Proportion

The sample proportion, denoted as \( \hat{p} \), is a crucial part of hypothesis testing. It is a way to measure how common a particular outcome is within a sample from a larger population. In simpler terms, it's like asking "Out of everyone we looked at, how many experienced a specific event?" For the hospital scenario, they're interested in the proportion of patients readmitted for pneumonia.To calculate \( \hat{p} \), use the formula:

• \( \hat{p} = \frac{\text{Number of successes}}{\text{Total number of trials}} \)

In the exercise, there were 11 readmissions out of 70 patients, so the sample proportion is \( \hat{p} = \frac{11}{70} \approx 0.1571 \). This means about 15.71% of patients in the sample were readmitted.Understanding the sample proportion helps set the stage for testing hypotheses about the population from which this sample was drawn.

Null Hypothesis

The null hypothesis, often represented as \( H_0 \), is the starting point for hypothesis testing. It is a statement asserting there is no effect or no difference. In other words, it suggests that any observed variation in the data is due to random chance and not a systematic effect.In the context of the pneumonia readmission rate, the null hypothesis might be:

• \( H_0: p = 0.17 \)

This hypothesis posits that the true proportion of readmissions at the hospital is the same as the national rate of 17%.The goal is to determine whether the evidence, based upon the sample data, is strong enough to reject this hypothesis.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis (\( H_a \)) reflects what the researcher aims to support. It suggests that the observed effect or difference is real and not just due to random chance.For our exercise, if the hospital believes their readmission rate is below the national average, the alternative hypothesis would be:

• \( H_a: p < 0.17 \)

This statement suggests that the hospital's rate is genuinely lower than 17%.The role of the alternative hypothesis is to provide a specific direction for the test, allowing researchers to identify significant trends in the data.

P-Value

The p-value is a fundamental concept in hypothesis testing and represents the probability of obtaining a test result at least as extreme as the one observed, assuming that the null hypothesis is true. It's a tool for deciding whether to reject the null hypothesis. In our example, a p-value of 0.39 was obtained. This means that there is a 39% chance of observing a sample proportion as low as 15.71% or lower, if the true readmission rate were actually 17%. Here's what that means:

• A high p-value (like 0.39) suggests that the observed data does not strongly contradict the null hypothesis.
• It indicates insufficient evidence to reject the null hypothesis at common significance levels such as 0.05 or 0.01.

Therefore, in practical terms, the evidence does not suggest the hospital's readmission rate is conclusively different from the national average.

Test Statistic

The test statistic is a standardized value that helps determine whether to reject the null hypothesis. It compares the sample data to what is expected under the null hypothesis.The formula for a test statistic when dealing with proportions is:

• \( Z = \frac{\hat{p} - p}{\sqrt{\frac{p (1-p)}{n}}} \)

Where:

• \( \hat{p} \) is the sample proportion
• \( p \) is the population proportion under the null hypothesis
• \( n \) is the sample size

Plugging in the values from the exercise: \( \hat{p} = 0.1571 \), \( p = 0.17 \), and \( n = 70 \), we calculate \( Z \approx -0.8641 \).This negative Z-value suggests that the sample proportion \( \hat{p} \) is less than the population proportion \( p \), underlining the direction suggested by the alternative hypothesis. However, the decision to reject the null hypothesis depends on comparing this Z-value to a critical value or using the p-value.