Null Hypothesis

A null hypothesis is a type of hypothesis used in statistics that proposes that no statistical significance exists in a set of given observations. The null hypothesis attempts to show that no variation exists between variables or that a single variable is no different than its mean. It is presumed to be true until statistical evidence nullifies it for an alternative hypothesis.

For example, if the hypothesis test is set up so that the alternative hypothesis states that the population parameter is not equal to the claimed value. Therefore, the cook time for the population mean is not equal to 12 minutes; rather, it could be less than or greater than the stated value. If the null hypothesis is accepted or the statistical test indicates that the population mean is 12 minutes, then the alternative hypothesis is rejected. And vice-versa.

The null hypothesis, also known as the conjecture, assumes that any kind of difference or significance you see in a set of data is due to chance. The opposite of the null hypothesis is known as the alternative hypothesis.

The null hypothesis is the initial statistical claim that the population mean is equivalent to the claimed. For example, assume the average time to cook a specific brand of pasta is 12 minutes. Therefore, the null hypothesis would be stated as, "The population mean is equal to 12 minutes." Conversely, the alternative hypothesis is the hypothesis that is accepted if the null hypothesis is rejected.

Hypothesis testing allows a mathematical model to validate or reject a null hypothesis within a certain confidence level. Statistical hypotheses are tested using a four-step process. The first step is for the analyst to state the two hypotheses so that only one can be right. The next step is to formulate an analysis plan, which outlines how the data will be evaluated. The third step is to carry out the plan and physically analyze the sample data. The fourth and final step is to analyze the results and either accept or reject the null hypothesis.

Here is a simple example: A school principal reports that students in her school score an average of 7 out of 10 in exams. To test this “hypothesis,” we record marks of say 30 students (sample) from the entire student population of the school (say 300) and calculate the mean of that sample. We can then compare the (calculated) sample mean to the (reported) population mean and attempt to confirm the hypothesis.

Take another example: the annual return of a particular mutual fund is 8%. Assume that mutual fund has been in existence for 20 years. We take a random sample of annual returns of the mutual fund for, say, five years (sample) and calculate its mean. We then compare the (calculated) sample mean to the (claimed) population mean to verify the hypothesis.

Usually the reported value (or the claim statistics) is stated as the hypothesis and presumed to be true. For the above examples, hypothesis will be:

• Example A: Students in the school score an average of 7 out 10 in exams.
• Example B: Annual return of the mutual fund is 8% per annum.

This stated description constitutes the “Null Hypothesis (H₀)” and is assumed to be true – the way a defendant in a jury trial is presumed innocent until proven guilty by evidence presented in court. Similarly, hypothesis testing starts by stating and assuming a “null hypothesis,” and then the process determines whether the assumption is likely to be true or false.

The important point to note is that we are testing the null hypothesis because there is an element of doubt about its validity. Whatever information that is against the stated null hypothesis is captured in the Alternative Hypothesis (H₁). For the above examples, the alternative hypothesis would be:

• Students score an average that is not equal to 7.
• The annual return of the mutual fund is not equal to 8% per annum.

In other words, the alternative hypothesis is a direct contradiction of the null hypothesis.

As an example related to financial markets, assume Alice sees that her investment strategy produces higher average returns than simply buying and holding a stock. The null hypothesis claims that there is no difference between the two average returns, and Alice has to believe this until she proves otherwise. Refuting the null hypothesis would require showing statistical significance, which can be found using a variety of tests. Therefore, the alternative hypothesis would state that the investment strategy has a higher average return than a traditional buy-and-hold strategy.

The p-value is used to determine the statistical significance of the results. A p-value that is less than or equal to 0.05 is usually used to indicate whether there is strong evidence against the null hypothesis. If Alice conducts one of these tests, such as a test using the normal model, and proves that the difference between her returns and the buy-and-hold returns is significant, or the p-value is less than or equal to 0.05, she can then refute the null hypothesis and accept the alternative hypothesis.