### What Is P-Value?

In statistics, the p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct. It is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. The p-value is used as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value means that there is stronger evidence in favor of the alternative hypothesis.

### How Is P-Value Calculated?

P-values are calculated using p-value tables or spreadsheets/statistical software. Because different researchers use different levels of significance when examining a question, a reader may sometimes have difficulty comparing results from two different tests.

For example, if two studies of returns from two particular assets were undertaken using two different significance levels, a reader could not compare the probability of returns for the two assets easily.

For ease of comparison, researchers often feature the p-value in the hypothesis test and allow the reader to interpret the statistical significance themselves. This is called a p-value approach to hypothesis testing.

### P-Value Approach to Hypothesis Testing

The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. The null hypothesis, also known as the conjecture, is the initial claim about a population of statistics.

The alternative hypothesis states whether the population parameter differs from the value of the population parameter stated in the conjecture. In practice, the p-value, or critical value, is stated in advance to determine how the required value to reject the null hypothesis.

### Type I Error

A type I error is the false rejection of the null hypothesis. The probability of a type I error occurring or rejecting the null hypothesis when it is true is equivalent to the critical value used. Conversely, the probability of accepting the null hypothesis when it is true is equivalent to 1 minus the critical value.

### Real-World Example of P-Value

Assume an investor claims that their investment portfolio's performance is equivalent to that of the Standard & Poor's (S&P) 500 Index. To determine this, the investor conducts a two-tailed test. The null hypothesis states that the portfolio's returns are equivalent to the S&P 500's returns over a specified period, while the alternative hypothesis states that the portfolio's returns and the S&P 500's returns are not equivalent. If the investor conducted a one-tailed test, the alternative hypothesis would state that the portfolio's returns are either less than or greater than the S&P 500's returns.

One commonly used p-value is 0.05. If the investor concludes that the p-value is less than 0.05, there is strong evidence against the null hypothesis. As a result, the investor would reject the null hypothesis and accept the alternative hypothesis.

Conversely, if the p-value is greater than 0.05, that indicates that there is weak evidence against the conjecture, so the investor would fail to reject the null hypothesis. If the investor finds that the p-value is 0.001, there is strong evidence against the null hypothesis, and the portfolio's returns and the S&P 500's returns may not be equivalent.