Question

Pairs of \(P\) -values and significance levels, \(\alpha,\) are given. For each pair, state whether the observed \(P\) -value leads to rejection of \(H_{0}\) at the given significance level. a. \(\quad P\) -value \(=.084, \alpha=.05\) b. \(\quad P\) -value \(=.003, \alpha=.001\) c. \(P\) -value \(=.498, \alpha=.05\) d. \(\quad P\) -value \(=.084, \alpha=.10\) e. \(\quad P\) -value \(=.039, \alpha=.01\) f. \(P\) -value \(=.218, \alpha=.10\)

Short answer

a. Fail to reject \(H_{0}\) \n b. Fail to reject \(H_{0}\) \n c. Fail to reject \(H_{0}\) \n d. Reject \(H_{0}\) \n e. Fail to reject \(H_{0}\) \n f. Fail to reject \(H_{0}\)

Step-by-step solution

Case a: Comparing P-value and Significance Level

Given that P-value = 0.084 and \(\alpha\) = 0.05. Since 0.084 is greater than 0.05, we fail to reject the null hypothesis \(H_{0}\).

Case b: Comparing P-value and Significance Level

Given that P-value = 0.003 and \(\alpha\) = 0.001. Since 0.003 is greater than 0.001, we fail to reject the null hypothesis \(H_{0}\).

Case c: Comparing P-value and Significance Level

Given that P-value = 0.498 and \(\alpha\) = 0.05. Since 0.498 is greater than 0.05, we fail to reject the null hypothesis \(H_{0}\).

Case d: Comparing P-value and Significance Level

Given that P-value = 0.084 and \(\alpha\) = 0.1. Since 0.084 is less than 0.1, we reject the null hypothesis \(H_{0}\).

Case e: Comparing P-value and Significance Level

Given that P-value = 0.039 and \(\alpha\) = 0.01. Since 0.039 is greater than 0.01, we fail to reject the null hypothesis \(H_{0}\).

Case f: Comparing P-value and Significance Level

Given that P-value = 0.218 and \(\alpha\) = 0.1. Since 0.218 is greater than 0.1, we fail to reject the null hypothesis \(H_{0}\).

Key concepts

Understanding the P-Value

The p-value is a crucial concept in hypothesis testing. It helps us make decisions about the null hypothesis. So, what exactly is a p-value? To put it simply, the p-value indicates the probability of obtaining test results at least as extreme as the observed data, assuming that the null hypothesis is true.
For example, if we have a p-value of 0.084, it means there is an 8.4% chance of observing data as extreme as what we have if the null hypothesis \(H_0\) is true.

• A small p-value (typically ≤ 0.05) suggests that the observed data is unlikely under the null hypothesis, leading us to consider rejecting \(H_0\).
• A large p-value (> 0.05) suggests that the observed data is consistent with the null hypothesis, meaning we fail to reject \(H_0\).

Significance Level Explored

The significance level, often denoted as \( \alpha \), is a threshold researchers set for determining whether the p-value is small enough to reject the null hypothesis. It's the risk level one is willing to take to reject \(H_0\) when it is actually true. Typically, common values for \( \alpha \) are 0.05, 0.01, or 0.10.
This means that:

• An \( \alpha\) of 0.05 indicates a 5% risk of incorrectly rejecting the null hypothesis.
• If our p-value is less than \( \alpha \), it suggests strong evidence against \(H_0\), so we reject it.
• If our p-value is greater than \( \alpha \), we do not have sufficient evidence to reject \(H_0\), so we fail to reject it.

The choice of \( \alpha\) depends on the field of study or specific experimental conditions. For high-stakes decisions, like drug approvals, researchers may choose a very low \( \alpha\), such as 0.01, to minimize the risk of incorrect rejection of \(H_0\).

The Role of the Null Hypothesis

The null hypothesis, denoted as \(H_0\), is the starting assumption for any hypothesis test. It often represents a position of no effect or no difference. For instance, if you're testing whether a new drug is effective, \(H_0\) might state that the drug has no effect.

In hypothesis testing, our goal is typically to see if we have enough evidence to reject \(H_0\) in favor of an alternative hypothesis \(H_1\), which represents an effect or difference. But we need substantial evidence to make such a decision.

• We always assume \(H_0\) is true until evidence suggests otherwise.
• The p-value shows how compatible your data is with \(H_0\).
• Failing to reject \(H_0\) doesn't prove \(H_0\) is true; it only shows that we lack strong evidence against it given our data.

It's important to remember that not rejecting \(H_0\) because of a higher p-value might simply mean we need more data or a different testing setup to make definitive conclusions.