Hypothesis testing is a method of statistical inference used to test a belief or claim about an unknown but fixed parameter *θ* of a population, using observed data from a subset of the population. By performing a *hypothesis test* on the measured sample, a statement can be made about the general population parameter *θ* by comparing it to the calculated sample parameter. Hypothesis testing is one of the most commonly applied models of probability and statistical theory, which many institutions use to make decisions to improve business operations. This report explains the method of constructing a hypothesis test and examines some of its applications in finance.

Constructing a hypothesis test is a methodological process. Consider a number *r* which is believed to be the value of the population parameter *θ*, or is a value against which is used to compare the value *θ*. Two hypotheses can be constructed, depending on what type of test will be conducted:

Null Hypothesis – H_{0} : *θ* = *r*

Alternative Hypothesis – H_{a} : *θ* ≠ *r ***or ***θ <* *r or θ > r*

Then consider the statistical assumptions being made and evaluate them in context with the underlying population; one of the most important factors to check for is the evidence of normality within the sample data. This can be checked using a Normal Quantile Plot. Determining the appropriate probability distribution of the sample follows from what characteristics are known about the general population. In most cases, the central limit theorem states that this is converges to the standard normal distribution, despite the individual distribution of each result within the sample. Hence the sample mean is distributed x̄~N(μ,σ^2/n), where μ, σ and n represent the population mean, standard deviation and sample size, respectively. In other cases where σ is unknown and the sample size n is small, a *t* *distribution* is better suited. The next value required is a *level of significance *α. This will determine the level of certainty at which the Null hypothesis can be rejected. The level of significance can be chosen to assess how reliable the conclusion of the test will be. Using the observed sample data, a *p-value *can then be found, which is the probability of observing a value at least as extreme as the sample statistic under the assumption that the Null Hypothesis is true. This value can then be compared to the significance level to draw a conclusion; if the p-value is smaller than the level of significance, the test is statistically significant, meaning there is enough evidence to reject H_{0 }at the α% level of significance.

In this way, hypothesis testing can be used extensively in confirming financial claims or ideas to a specific level of certainty. For example, it could be used to determine what an investor should invest in and to what extent a satisfactory return will be achieved. Given the previous behaviour of the return of investment (ROI) of a portfolio, a sample could be created from a subset of the portfolio data displaying the monthly incomes generated from the portfolio. If the average value of the income generated is known, along with its standard deviation, then p-values could be used to generate the probability that the subsequent months will provide a satisfactory return.

Hypothesis tests can also be tailored to the specific need of the business that conducts them. Low significance levels are desired as it means that the probability of incorrectly rejecting H_{0} is equally low (known as a *Type I error*). Also, a test is more reliable when the probability that H_{0 }is rejected when it is indeed false (known as a *Type II error*) is small. The *power *of the test is defined as 1 – P(Type II error | H_{a}). Hence an ideal test would have a low significance level and high power; often the hypothesis test that is conducted incorporates a trade-off between these two values.

Thus, the accumulation of all these pieces of information could persuade, or even dissuade, a potential investment. In either case, the use of mathematics and statistics at the heart of hypothesis testing has provided the opportunity to maximise a business’ own utility.

By Ayush Bose

Sector Head: Aaron Hobb