As a domain expert in statistical analysis, I would like to clarify the concept of a "negative p-value" and its implications in hypothesis testing. A p-value is a statistical measure that indicates the strength of the evidence against the null hypothesis. It is calculated from the observed data and is used to determine whether the results of an experiment or study are statistically significant.
When we conduct a hypothesis test, we typically start with a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis usually represents a status quo or a claim that we are testing, while the alternative hypothesis represents the opposite of the null hypothesis or what we would like to show to be true.
The negative sign of a p-value does not inherently imply that the sample mean is less than the hypothesized mean. Instead, it indicates that the observed results are less likely to have occurred if the null hypothesis were true. The positive sign of a p-value would suggest the opposite—that the observed results are more likely under the null hypothesis.
Here's a step-by-step breakdown of how a negative p-value might come into play:


1. Formulate the Hypotheses: We start by defining the null hypothesis (H0) and the alternative hypothesis (H1). For example, H0 might state that there is no difference in the means of two groups (μ1 = μ2), while H1 might claim that the mean of one group is greater than the other (μ1 > μ2).


2. Collect and Analyze Data: We then collect data and perform a statistical test, such as a t-test or a z-test, to compare the means of the two groups.


3. Calculate the Test Statistic: The test statistic is calculated based on the sample data. This statistic follows a specific distribution under the null hypothesis.


4. Determine the P-Value: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from our sample data, assuming the null hypothesis is true.


5. Interpret the P-Value: If the p-value is less than a predetermined significance level (commonly 0.05), we reject the null hypothesis in favor of the alternative hypothesis. This is considered statistically significant.


6. Consider the Direction: The direction of the p-value (whether it is negative or positive) is not as important as its magnitude. What matters is whether the p-value is below the significance level. However, the direction can be informative if we are testing a one-tailed hypothesis. For a one-tailed test where H1 is μ1 > μ2, a negative test statistic (indicating the sample mean is less than the hypothesized mean) would not support the alternative hypothesis. Conversely, a positive test statistic would support it.

7.
Make a Decision: Based on the p-value, we make a decision to either reject or fail to reject the null hypothesis.

It's important to note that a p-value is not the probability that the null hypothesis is true or false. It is also not the probability that the alternative hypothesis is true. It is simply a measure of the strength of the evidence against the null hypothesis.

In conclusion, a negative p-value does not mean that the sample mean is less than the hypothesized mean; rather, it means that the observed data are unlikely under the assumption that the null hypothesis is true, and it provides evidence against the null hypothesis when the test is conducted in the correct direction. The interpretation of the p-value should always be made in the context of the specific hypotheses being tested and the directionality of the test.

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A negative sign implies that the sample mean is less than the hypothesized mean. ... A positive sign implies that the sample mean is larger than the hypothesized mean. This would be evidence against the null hypothesis IF (and only if) the alternative was that the true mean is GREATER than the hypothesized value.read more >>