Question

Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.007\) or \(\quad\) p-value \(=0.13\)

Short answer

A p-value of 0.007 provides the strongest evidence against the null hypothesis.

Step-by-step solution

Comparing P-values

p-value of 0.007 and p-value of 0.13 are compared. Since 0.007 is smaller than 0.13, it indicates stronger evidence against the null hypothesis.

Key concepts

Null Hypothesis

The null hypothesis, typically denoted as \(H_0\), is one of the core concepts in hypothesis testing. It represents the default or original claim that there is no effect or no difference in the context of the experiment or study being conducted. For instance, if researchers are testing a new drug, the null hypothesis might state that the drug has no effect on patients compared to a placebo.

In statistical terms, \(H_0\) serves as the assertion to be tested, and it's formulated in such a way that it allows for a clear decision to be made. The null hypothesis is paired with an alternative hypothesis, denoted as \(H_1\) or \(H_a\), which claims that there is indeed an effect or a difference. The objective of hypothesis testing is to make a decision about which hypothesis is supported by the sample data, keeping in mind that the test could potentially lead to a type I error (rejecting \(H_0\) when it's true) or a type II error (failing to reject \(H_0\) when it's false).

Evidence Against Null Hypothesis

Evidence against the null hypothesis arises in the form of statistically significant results, which can cast doubt on \(H_0\)'s validity. The primary tool used to measure this evidence is the p-value, which quantifies how likely the sample data is, assuming that the null hypothesis is true.

When the p-value falls below a pre-determined significance level, often set at 0.05, it suggests that the observed data would be very unlikely if the null hypothesis were true. This is considered strong evidence against \(H_0\), leading researchers to reject the null and consider the alternative hypothesis as more likely. It's important to understand that a low p-value doesn't prove that \(H_0\) is false or that \(H_1\) is true; it merely indicates that the sample data is inconsistent with what we would expect to see if \(H_0\) were correct.

Comparing P-values

Comparing p-values is an integral part of hypothesis testing, as it informs researchers which of the results presents the stronger evidence against the null hypothesis. A smaller p-value indicates that the observed effect is less likely to be due to random chance and therefore is more convincing evidence against \(H_0\).

In the context of the given exercise, a p-value of 0.007 is substantially smaller than a p-value of 0.13. This indicates that the evidence against the null hypothesis is stronger when the p-value is 0.007. Essentially, it means there's a 0.7% chance of the sample data occurring by random variation if \(H_0\) were true, compared to a 13% chance for the larger p-value. In practical applications, comparing p-values can help prioritize findings and determine which hypotheses are worth rejecting and investigating further.