Question

Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.90\) or \(\quad\) p-value \(=0.08\)

Short answer

The p-value of 0.08 provides stronger evidence against the null hypothesis \(H_{0}\) compared to the p-value of 0.90.

Step-by-step solution

Understand the meaning of p-value

The p-value is a measure of the probability that an observed data set would be obtained by random chance, given that the null hypothesis is true. If the p-value is small, it means the observed data is very unlikely under the null hypothesis, and thus it provides strong evidence against the null hypothesis

Compare the two given p-values

Having the p-values 0.90 and 0.08, it can be said that the p-value of 0.90 is larger compared to 0.08. In the terms of sufficiency of evidence against the null hypothesis, a small p-value is more sufficient than a larger one.

Determine which p-value provides stronger evidence against the null hypothesis

As the p-value of 0.08 is smaller than the p-value of 0.90, it can be concluded that the p-value of 0.08 provides stronger evidence against the null hypothesis \(H_{0}\).

Key concepts

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a fundamental concept in statistical hypothesis testing. It proposes that there is no significant difference or effect and that any observed variations are due to chance or random fluctuations. For example, if we were testing a new drug's effectiveness, the null hypothesis would assert that the drug has no effect on patients compared to a placebo.

When performing a hypothesis test, researchers aim to gather evidence that either supports or refutes the null hypothesis. A key aspect of this process is determining whether the data collected is significantly different from what would be expected if the null hypothesis were true. If the evidence strongly contradicts the null hypothesis, researchers may reject it in favor of an alternative hypothesis, which suggests that there is a significant effect or difference.

Probability

Probability is a measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). It’s an intrinsic part of statistics and plays a crucial role in hypothesis testing, where it helps to quantify the likelihood of obtaining the observed results under the assumption of the null hypothesis.

For instance, in the context of p-values, probability refers to the chance of observing data as extreme as, or more extreme than, the results obtained during the test, assuming that the null hypothesis is true. This concept allows researchers to make quantitative decisions about the data. It’s essential to remember that while probability can indicate how likely an event is, it is not a definitive prediction. A low probability does not guarantee that an event won't occur; it just means it's less likely.

Statistical Significance

Statistical significance is a determination about whether the observed results in a study or experiment are unlikely to have occurred by chance. This determination is typically made by calculating the p-value during hypothesis testing. The p-value measures the strength of the evidence against the null hypothesis.

A common threshold for declaring statistical significance is a p-value of less than 0.05, meaning there is less than a 5% probability that the observed results occurred by random chance, considering the null hypothesis is true. When a result is statistically significant, it suggests that the observed effect or difference is likely real and not just a random occurrence. However, it's important to note that statistical significance does not imply practical significance; the result may be statistically reliable but not necessarily large or important in a practical sense.