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Chapter 4: Problem 41

In Exercises 4.41 to 4.44 , two p-values are given. Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.90\) or \(\quad\) p-value \(=0.08\)

Short Answer

Expert verified
The p-value equals 0.08 provides the strongest evidence against the null hypothesis (H0).

Step by step solution

01

Understanding the p-value in hypothesis testing

In hypothesis testing, the p-value is used as a tool to decide the fate of the null hypothesis. If the p-value is small, usually less than or equal to a specified significance level (often denoted by 伪), then strong evidence against the null hypothesis is provided and it is rejected in favor of the alternative hypothesis. The smaller the p-value, the less likely the results are under the null hypothesis, hence the stronger the evidence against H0.
02

Comparing given p-values

Here, two p-values are given: 0.90 and 0.08. Remember from the Step 1 that the lower the p-value, the stronger the evidence against the null hypothesis.
03

Determining the strongest evidence against H0

The p-value of 0.08 is smaller than the p-value of 0.90. Thus, p-value=0.08 provides the strongest evidence against the null hypothesis (H0).

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Key Concepts

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

p-value analysis
The concept of p-value in hypothesis testing plays a crucial role in deciding whether to reject the null hypothesis. It helps quantify the probability of observing results at least as extreme as those obtained, under the assumption that the null hypothesis is true. A p-value reflects the strength of evidence against the null hypothesis:
  • A smaller p-value indicates stronger evidence against the null hypothesis. It suggests that the observed data is less likely to occur if the null hypothesis is true.
  • If the p-value is smaller than the predetermined significance level (often 0.05), the null hypothesis is typically rejected.
  • The comparison of different p-values allows researchers to interpret which scenario is less likely under the null hypothesis.
For example, a p-value of 0.08 implies stronger evidence against the null hypothesis compared to a p-value of 0.90.
null hypothesis
In statistics, the null hypothesis, denoted as \( ext{H}_0\), serves as the default or initial claim that there is no effect or no difference. It often represents the idea that "nothing is happening" or that observed effects are due to chance. When conducting hypothesis testing:
  • The null hypothesis posits that any kind of difference or significance you notice is purely coincidental and not necessarily caused by any outside factors you are measuring.
  • Researchers attempt to challenge the null hypothesis by providing evidence through experiments or sampling that shows it may not be true.
  • The primary objective is to determine whether there is enough statistical evidence to reject the null hypothesis in favor of the alternative hypothesis.
Thus, a hypothesis test is essentially a method to evaluate whether \( ext{H}_0\) should be rejected or not based on the data.
significance level
The significance level, commonly denoted by the Greek letter \( \alpha \), is a threshold set by researchers to determine when to reject the null hypothesis. It outlines how much risk we are willing to accept in terms of mistakenly rejecting a true null hypothesis. Here鈥檚 how it works:
  • The significance level is typically set at 0.05, although it can be lower (such as 0.01) or higher, depending on the field of study or specific use case.
  • A smaller \( \alpha \) signifies a stricter requirement for evidence against the null hypothesis, reducing the risk of a Type I error (false positive).
  • When a calculated p-value is less than or equal to \( \alpha \), it suggests the data provides sufficient evidence to reject the null hypothesis. Conversely, if the p-value is greater than \( \alpha \), the null hypothesis is not rejected.
In hypothesis testing, choosing the right significance level is critical as it influences the conclusions drawn about the null hypothesis.

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Most popular questions from this chapter

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