###### Tracking Error

Tracking error refers to the difference in returns between a portfolio (index fund)... **Read More**

A hypothesis is an assumptive statement about a problem, idea, or some other characteristic of a population. It can also be considered an opinion or claim about a given issue. Therefore, a statistical test has to be performed to establish whether a hypothesis is correct or not. Hypothesis testing involves using sample data to assess if a sample statistic represents a population with the hypothesized value of the population parameter.

Below is an example of a hypothesis:

“The mean lifetime of men is less than that of women.”

Hypothesis testing involves the collection and examination of a representative sample to verify the accuracy of a hypothesis. Hypothesis tests help analysts to answer questions such as:

- Is bond type A more profitable than type B?
- Does staff training lead to improved efficiency at the workplace?
- Are motor vehicle insurance claims consistent with a lognormal distribution?

Whenever a statistical test is being performed, the following procedure is generally considered ideal:

- Statement of both the null and the alternative hypotheses.
- Selection of the appropriate test statistic, i.e., what’s being tested, e.g., the population mean, the difference between sample means, or variance.
- Specification of the level of significance.
- Clear statement of the decision rule to guide the choice of whether to reject or approve the null hypothesis.
- Calculation of the sample statistic.
- Arrival at a decision based on the sample results.

The **null hypothesis**, denoted as H_{0}, represents the current state of knowledge about the population parameter that is the subject of the test. In other words, it represents the “status quo.” For example, the U.S Food and Drug Administration may walk into a cooking oil manufacturing plant intending to confirm that, indeed, the cholesterol content of each 1 kg oil package does not exceed, say, 0.15%. The inspectors will formulate a hypothesis like:

H_{0 } \(\le \) Each 1 kg package has 0.15% cholesterol.

A test would then be carried out to confirm or reject the null hypothesis.

Typical statements of H_{0 }include:

$$ H_0: \ \mu = \mu_0 $$

$$ H_0: \ \mu \le \mu_0 $$

$$ H_0: \ \mu \ge \mu_0 $$

Where:

μ = True population mean.

μ_{0 }= Hypothesized population mean.

The **alternative hypothesis**, denoted as H_{1}, is a contradiction of the null hypothesis. Therefore, rejecting the H_{0} makes H_{1 }valid. We accept the alternative hypothesis when the “status quo” is discredited and found to be false.

Using our FDA example above, the alternative hypothesis would be:

H_{1}: Each 1 kg package does not have 0.15% cholesterol.