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Chapter 4: Problem 21

Exercises 4.21 to 4.25 describe tests we might conduct based on Data 2.3 , introduced on page \(66 .\) 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 evidence that mean heart rate is higher in male ICU patients than in female ICU patients?

Short Answer

Expert verified
The null hypothesis is \( H_{0} : \mu_{M} = \mu_{F} \), asserting there is no difference in mean heart rates between male and female ICU patients. The alternative hypothesis is \( H_{A} : \mu_{M} > \mu_{F} \), contending that the mean heart rate is higher in male ICU patients than in female patients.

Step by step solution

01

Identify the relevant parameters

The relevant parameters in this case are the mean heart rates of male and female ICU patients. We denote the mean heart rate of male ICU patients as \( \mu_{M} \) and that of female ICU patients as \( \mu_{F} \).
02

State the null hypothesis

The null hypothesis, denoted by \( H_{0} \), is generally a statement of no effect or no difference. In the context of this research question, it means there is no difference between the heart rates of male and female ICU patients. Mathematically, this can be stated as \( H_{0} : \mu_{M} = \mu_{F} \).
03

State the alternative hypothesis

The alternative hypothesis, denoted by \( H_{A} \), is a statement that contradicts the null hypothesis. It embodies the assertion we are trying to prove. In this case, we are trying to prove that the mean heart rate is higher in male ICU patients. Thus the alternative hypothesis can be stated as \( H_{A} : \mu_{M} > \mu_{F} \).

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Key Concepts

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

Understanding the Null Hypothesis
In statistical hypothesis testing, the **null hypothesis** is the cornerstone. It's a common starting point that often represents the idea of "no effect" or "no difference." When you formulate the null hypothesis, you are assuming that any kind of differences or patterns you see in your data are completely due to chance.

For the ICU Admissions example, the null hypothesis (\( H_0 \)) states that there is no difference in the mean heart rates of male and female ICU patients. This means that any observed difference in heart rates could be just random variability.

The expression for this hypothesis is:
\( H_{0} : \mu_{M} = \mu_{F} \)
where \( \mu_{M} \) and \( \mu_{F} \) are the mean heart rates of male and female patients, respectively. The null hypothesis is tested to see if there is enough statistical evidence to reject it, thereby providing support for the alternative hypothesis.
Exploring the Alternative Hypothesis
The **alternative hypothesis** is like the null hypothesis's counterpart. It suggests that there is an effect or a difference that goes beyond random chance. This hypothesis is what you aim to support through your data and analysis.

Regarding our exercise, the alternative hypothesis (\( H_A \)) asserts that there is indeed a difference, such as a higher mean heart rate in male patients compared to females. It's focused on detecting a specific direction of effect.

Mathematically, it is expressed as:
\( H_{A} : \mu_{M} > \mu_{F} \)
This notation indicates that the mean heart rate for male ICU patients (\( \mu_{M} \)) is greater than that for female patients (\( \mu_{F} \)).

In hypothesis testing, if you find sufficient statistical evidence that supports the alternative hypothesis, then the null is rejected. This is the goal of the test: to provide evidence for an effect that the alternative hypothesis claims.
  • The alternative hypothesis challenges the status quo.
  • It seeks to show that any observed difference is significant.
Clarifying Statistical Parameters
When dealing with hypothesis testing, **statistical parameters** play a critical role as they define the characteristics or properties you're interested in. Parameters are essential since they inform your hypotheses and the statistic computations.

In the context of the ICU study, we're looking at parameters like the mean heart rates. These are represented as \( \mu_{M} \) for male patients and \( \mu_{F} \) for female patients. They're essentially the averages that you calculate during your investigation.

Understanding these parameters helps you construct both null and alternative hypotheses accurately because they set the population metrics you wish to compare or analyze.

For your hypothesis tests:
  • Parameters define what aspect of your data you're testing.
  • They provide a basis to apply statistical models and interpret outcomes.
Careful definition and comprehension of statistical parameters are necessary to ensure your hypothesis testing process is both valid and reliable.

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Most popular questions from this chapter

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.7\) vs \(H_{a}: p<0.7\) Sample data: \(\hat{p}=125 / 200=0.625\) with \(n=200\)

Testing for a Gender Difference in Compassionate Rats In Exercise 3.80 on page 203 , we found a \(95 \%\) confidence interval for the difference in proportion of rats showing compassion, using the proportion of female rats minus the proportion of male rats, to be 0.104 to \(0.480 .\) In testing whether there is a difference in these two proportions: (a) What are the null and alternative hypotheses? (b) Using the confidence interval, what is the conclusion of the test? Include an indication of the significance level. (c) Based on this study would you say that female rats or male rats are more likely to show compassion (or are the results inconclusive)?

In Exercises 4.112 to \(4.116,\) the null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2}\) Sample: \(\bar{x}_{1}=2.7\) and \(\bar{x}_{2}=2.1\) Randomization statistic \(=\bar{x}_{1}-\bar{x}_{2}\)

4.20 Taste Test A taste test is conducted between two brands of diet cola, Brand \(A\) and Brand \(B\), to determine if there is evidence that more people prefer Brand A. A total of 100 people participate in the taste test. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Give an example of possible sample results that would provide strong evidence that more people prefer Brand A. (Give your results as number choosing Brand \(\mathrm{A}\) and number choosing Brand B.) (c) Give an example of possible sample results that would provide no evidence to support the claim that more people prefer Brand A. (d) Give an example of possible sample results for which the results would be inconclusive: The sample provides some evidence that Brand \(\mathrm{A}\) is preferred but the evidence is not strong.

Classroom Games Exercise 4.62 describes a situation in which game theory students are randomly assigned to play either Game 1 or Game 2 , and then are given an exam containing questions on both games. Two one-tailed tests were conducted: one testing whether students who played Game 1 did better than students who played Game 2 on the question about Game \(1,\) and one testing whether students who played Game 2 did better than students who played Game 1 on the question about Game \(2 .\) The p-values were 0.762 and 0.549 , respectively. The p-values greater than 0.5 mean that, in the sample, the students who played the opposite game did better on each question. What does this study tell us about possible effects of actually playing a game and answering a theoretical question about it? Explain.
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