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Chapter 4: Problem 22

Describe tests we might conduct based on Data 2.3 , introduced on page \(69 .\) This dataset, stored in ICUAdmissions, contains information about a sample of patients admitted to a hospital Intensive Care Unit (ICU). For each of the research questions below, define any relevant parameters and state the appropriate null and alternative hypotheses. Is there a difference in the proportion who receive CPR based on whether the patient's race is white or black?

Short Answer

Expert verified
The relevant parameters are the proportions of white patients (\( p_{1} \)) and black patients (\( p_{2} \)) who receive CPR. The Null Hypothesis is \( H_{0} : p_{1} = p_{2} \) implying there is no difference in the proportions of white and black patients receiving CPR. The Alternative Hypothesis is \( H_{1} : p_{1} 鈮� p_{2} \) implying there is a difference. The appropriate test is a two-proportion z test.

Step by step solution

01

Understanding the Task

Data 2.3, consists of data about various patients admitted to a hospital's ICU. The task attempts to find out if there is a difference in the proportion of patients who receive CPR based on their race - white or black.
02

Identify Parameters

Parameters are important quantities in populations that we want to estimate or test. In this case, the parameters are the proportions of white and black patients who receive CPR. Let's use \( p_{1} \) to represent the proportion of white patients who receive CPR and \( p_{2} \) to represent the proportion of black patients who receive CPR.
03

State the Null Hypothesis

The Null Hypothesis usually states there is no effect or no difference in our case of comparison. Thus our null hypothesis will be: \( H_{0} : p_{1} = p_{2} \) which means there is no difference in the proportions of white and black patients receiving CPR.
04

State the Alternative Hypothesis

The Alternative Hypothesis is the contrary to what the null hypothesis assumes. It often states there is an effect or a difference. Thus our alternative hypothesis would be \( H_{1} : p_{1} 鈮� p_{2} \), indicating there is a difference in the proportions of white and black patients receiving CPR.
05

Chose Appropriate Test

We have two populations (white and black patients) and we are comparing their proportions (those who receive CPR). The appropriate test in this scenario would be a two-proportion z test.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis is a critical concept. It represents an initial claim that there is no effect or no difference in a situation. When we are comparing two groups, like in our case with ICU admissions, the null hypothesis is a starting point. It helps us to determine if any observed differences are due to random chance or an actual effect.

For instance, in our ICU case, we are interested in whether there is a difference in the proportions of white and black patients receiving CPR. The null hypothesis is expressed as:
  • \( H_{0} : p_{1} = p_{2} \)
This means we assume there is no difference in the CPR receiving proportions between the two racial groups.

When conducting the test, we collect data and analyze it statistically to decide whether to 'reject' or 'fail to reject' this hypothesis. If we find significant evidence against it, it implies that there might indeed be a notable difference between the groups.
Alternative Hypothesis
The alternative hypothesis is the counterpart to the null hypothesis. Instead of assuming no difference or no effect, it proposes what you expect to find in your study - a difference or effect.

In our ICU example, if the null hypothesis states that the proportion of white and black patients receiving CPR is the same, the alternative hypothesis suggests otherwise:
  • \( H_{1} : p_{1} eq p_{2} \)
Here, \( H_{1} \), indicates that there is a difference between the proportions of the two groups.

The ultimate goal of the experiment or study is to collect enough statistical evidence to support this alternative hypothesis. If the data supports \( H_{1} \), we uncover insights, suggesting an actual discrepancy in the proportions between the groups.
Two-Proportion Z Test
The Two-Proportion Z Test is a statistical method used to determine if there is a significant difference between the proportions of two groups. This test is particularly useful when comparing two populations in terms of a single characteristic or event.

In our ICU admissions problem, the two-proportion z test helps us figure out if the proportions of white versus black patients receiving CPR are different enough to be considered statistically significant. When conducting this test, we:
  • Calculate the estimated proportions of the groups receiving CPR.
  • Use a formula to find the z-statistic, which tells us how much the observed difference between proportions deviates from what we would expect if \( p_{1} = p_{2} \).
  • Compare the z-statistic to a critical value from the standard normal distribution to decide whether to reject the null hypothesis.

A significant result would suggest that the difference in proportions is unlikely to be due to chance alone, thus giving more weight to the alternative hypothesis.

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Most popular questions from this chapter

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