What is a null hypothesis?
The null hypothesis constitutes an statistical proposition that suggests the absence of statistical significance in the analyzed observations. It is applied in hypothesis testing to determine the plausibility of a hypothesis using data obtained from samples. Commonly referred to as “null”, it is symbolized by H0.
This conjecture is often used in quantitative analyzes to examine theories relating to markets, investment strategies and economic models, with the aim of verifying the veracity of certain assumptions.
How does a null hypothesis work?
The null hypothesis represents an assumption in statistics that postulates the non-existence of significant differences between specific characteristics of a population or a data generation process.
For example, a bettor may want to check whether a game is fair. If the game is fair, the expected winnings for each participant should be zero. Otherwise, the expected gains will be positive for one player and negative for the other player. To ascertain the fairness of the game, the bettor collects data from multiple rounds, calculates the average winnings, and tests the null hypothesis that the expected winnings are equivalent to zero.
If the sample average earnings differ significantly from zero, the bettor will reject the null hypothesis, accepting the alternative hypothesis that the expected earnings differ from zero. If the average earnings approach zero, the bettor will maintain the null hypothesis, attributing the observed difference to chance.
The null hypothesis assumes that discrepancies observed in the data are the result of random variations. For example, if actual earnings are zero, any deviation of average earnings from zero is due to chance.
Analysts try to refute the null hypothesis as it indicates robust evidence. This requires compelling evidence, such as an observed discrepancy that is too large to be attributed to chance. Failure to reject the null hypothesis indicates a weaker conclusion, allowing factors beyond chance to influence the results, although they are not detected by the statistical test because they are not strong enough.
The alternative hypothesis
It is crucial to note that null hypothesis testing is conducted due to the existence of uncertainty regarding its validity. Any data that contradicts the null hypothesis is considered when formulating the alternative hypothesis (H1).
For example, alternative hypotheses for the mentioned cases would be:
- Students have a different average of seven.
- The average annual mutual fund return is not equal to 8% per year.
The alternative hypothesis, therefore, directly refutes the null hypothesis.
Examples of a null hypothesis
Consider the following case: the director of an educational institution claims that the average grade of students is seven out of ten. The null hypothesis formulated would be that the average of the student population is 7,0. To verify this proposition, one could record the grades of, for example, 30 students out of a total of 300 and calculate the average of these collected data.
From there, the mean obtained from the sample would be compared to the hypothetical population mean of 7,0 to assess the possibility of rejecting the null hypothesis. It is important to emphasize that the null hypothesis, in this context — that the population mean is 7,0 — cannot be directly confirmed by the sample data, only rejected.
In another scenario, the annual return of a specific mutual fund is assumed to be 8%. Suppose this fund has operated for twenty years. Here, the null hypothesis would be that the average annual return is 8%. A random sample of annual returns over five years could be examined to calculate the sample mean. This average would then be compared to the stated average return of 8% to test the null hypothesis.
In these examples, the null hypotheses are:
- Example A: Students achieve an average of seven out of ten on exams.
- Example B: The fund's average annual return is 8%.
To determine the feasibility of rejecting the null hypothesis, it is initially assumed that it is true to establish a likely range of possible values for the calculated statistic. For example, the range of acceptable means can vary from 6,2 to 7,8, considering a population mean of 7,0. If the sample mean is outside this range, the null hypothesis is rejected. Otherwise, the variation is considered to be explainable solely by chance.
How null hypothesis testing is used in investing
Take the hypothetical case of Alice, who sees higher average returns with her investment strategy compared to buying and holding a stock. The null hypothesis here would be that there is no difference between the average returns of the two approaches. Alice maintains this belief until contrary evidence is sufficiently substantial.
To refute the null hypothesis, it would be necessary to demonstrate statistical significance, which can be achieved through several tests. The alternative hypothesis would suggest that Alice's investment strategy provides a higher average return than the conventional buy-and-hold strategy.
A crucial element in this analysis is the p-value, which indicates the probability that an observed difference is the result of chance alone. A p-value of 0,05 or less generally suggests sufficient evidence against the null hypothesis.
If Alice applies one of these statistical tests, such as one based on the normal model, and obtains a p-value that indicates a significant difference between her returns and those of the buy-and-hold strategy, she can then reject the null hypothesis and adopt the alternative hypothesis.
Conclusion
Understanding and applying the null and alternative hypotheses are fundamental to statistical research in several areas, including finance. The null hypothesis serves as a starting point for statistical tests, offering a standard assumption that there is no effect or significant difference in a given set of data. On the other hand, the alternative hypothesis offers a contrary perspective, which is validated or refuted through rigorous analysis.
The process of testing these hypotheses is meticulous and requires careful planning, precise execution, and careful interpretation of results. This method not only clears up doubts about the validity of a theory or strategy, but also drives informed, evidence-based decision-making, particularly in sectors such as finance, where decisions can have significant implications.
FAQ
How is the null hypothesis identified?
Identification of the null hypothesis occurs when the analyst or researcher defines an initial assumption based on the problem or research question he or she wishes to investigate. The nature of the null hypothesis varies depending on the specific question under analysis. For example, if the question investigates the existence of an effect (such as, does X influence Y?), the null hypothesis would be H0: suggest that the effect of X on Y is positive, then H0 would be X > 0. Rejection of the null hypothesis occurs if the results of the analysis show a statistically significant effect that differs from zero.
How is the null hypothesis used in finance?
In the financial sector, the null hypothesis is often applied in quantitative analyzes to assess the validity of investment strategies, the behavior of markets or the performance of economies. A practical example would be an analyst trying to determine whether there is a significant correlation between the shares of two companies, ABC and XYZ. The null hypothesis in this case could be formulated as ABC ≠ XYZ.
How are statistical hypotheses tested?
Statistical hypothesis testing involves a structured process in four steps. Initially, the analyst formulates both hypotheses so that only one can be true. The second stage consists of preparing an analytical plan detailing the data evaluation methodology. This is followed by the execution of the plan and the effective analysis of the sample data. The final stage covers the interpretation of the results, where one can choose to reject the null hypothesis or conclude that the observed differences can be justified by chance.
What is an alternative hypothesis?
The alternative hypothesis represents the direct contradiction to the null hypothesis. Essentially, if one of the hypotheses is proven to be true, the other will automatically be considered false. This opposite positioning serves to facilitate statistical analysis, offering a clear counterpoint to the initial assumption examined.
