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Chapter 4: Problem 69

Exercises 4.67 to 4.70 give a p-value. State the conclusion of the test based on this p-value in terms of " Reject \(H_{0} "\) or "Do not reject \(H_{0} "\) if we use a \(5 \%\) significance level. $$ \text { p-value }=0.2531 $$

Short Answer

Expert verified
Do not reject \(H_{0}\)

Step by step solution

01

Compare the p-value to the significance level

Firstly, compare the given p-value to the significance level. The p-value for this test is 0.2531 and the given significance level (\(\alpha\)) is 0.05.
02

Making the decision to reject or not reject the null hypothesis

If the p-value is less than or equal to the significance level (\(0.2531 \leq 0.05\)), reject \(H_{0}\). If the p-value is more than the significance level (\(0.2531 > 0.05\)), do not reject \(H_{0}\). In this case, as 0.2531 is greater than 0.05, the conclusion is to not reject the null hypothesis.

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Key Concepts

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

Understanding the Significance Level
The significance level, often denoted by \( \alpha \), plays a pivotal role in hypothesis testing. It acts like a threshold that determines the degree of confidence you want in your results.
The most commonly used significance level is 5% or 0.05.
This means that there is a 5% risk of rejecting the null hypothesis when it is actually true. When you set a significance level:
  • You're deciding how much evidence you require to reject the null hypothesis.
  • You're balancing the cost of a Type I error (false positive) with finding an effect if there is one.
In practice, researchers choose a significance level based on the context and consequences of their decision. With a 5% level, you're usually aiming to confidently claim a result as significant, without too much risk of making a wrong decision.
Decoding the P-Value
The p-value is a fundamental concept in statistical testing. It tells you how probable it is to observe your data, or something more extreme, assuming the null hypothesis is true.
In simple terms, it provides a measure of the evidence against the null hypothesis. Here's how p-values work:
  • A small p-value (typically 鈮� 0.05) suggests rejecting the null hypothesis, indicating strong evidence against it.
  • A large p-value (> 0.05) implies weak evidence against the null, supporting not rejecting it.
In our example, the p-value was 0.2531. This value is significantly larger than the commonly used significance level of 0.05. Therefore, it suggests insufficient evidence to reject the null hypothesis.
Explaining the Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is a key part of hypothesis testing. It is a statement that implies no effect or no difference, and serves as a starting point for testing.Why is it so important?
  • It provides a baseline that can be tested statistically.
  • Allows you to determine if your observed data can be due to random chance.
When you're conducting a hypothesis test, your primary goal is to decide whether you should reject the null hypothesis. In many cases, rejecting \( H_0 \) suggests that there is an effect or difference that is statistically significant. In our case, the p-value was not small enough to reach this conclusion, so you would "Do not reject \( H_0 \)". This means the data did not provide strong enough evidence to support an alternative claim.

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Most popular questions from this chapter

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