Many a times, we want to test the validity of a statement. For example, is the mean return from this mutual fund more to the mean return from the benchmark? While answering such a question, our interest is not to find the actual mean returns of the mutual fund, but to test whether the statement holds true or not.

A hypothesis is a statement about one or more populations like the statement above.

A hypothesis testing is a standard procedure to test the hypothesis. There could be two possible results: 1) The hypothesis is correct and hence should be accepted. 2) The hypothesis is incorrect and should be rejected.

Steps in Hypothesis Testing

The hypothesis testing process consists of the following steps:

1. Stating the hypotheses.
2. Identifying the appropriate test statistic and its probability distribution.
3. Specifying the significance level.
4. Stating the decision rule.
5. Collecting the data and calculating the test statistic.
6. Making the statistical decision.

Null and Alternative Hypothesis

When we form a hypothesis to be tested, the hypothesis is called a null hypothesis. The null hypothesis is written as H0. A null hypothesis will be a simple statement about the population parameter. For example, the hypothesis that the mean returns of a mutual fund will be greater than or equal to 8% will be states as H0: m >= 8%.

The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. It is represented as HA. In our example, the alternative hypothesis will be HA: m < 8%.

In general, a hypothesis can be formed as follows:

1. H0: θ = θ0 versus HA: θ ≠ θ0
2. H0: θ ≤ θ0 versus HA: θ > θ0
3. H0: θ ≥ θ0 versus HA: θ < θ0

Where θ represents a population parameter and θ0 is a value assumed in the hypothesis.

One-tailed and Two-tailed Tests

A hypothesis test can be a one-tailed or a two-tailed test. A one-tailed test means that the hypothesis is one-sided such as the second and third formulation above. The second formulation tests whether the population parameter is greater than a certain value (one-sided). The third formulation tests whether the population parameter is greater than a certain value (again one-sided). The first formulation is two-sided; hence the hypothesis test will be twp-tailed. It tests for deviation of value on both sides of θ0.

Example

We will use this example to illustrate the concepts related to hypothesis testing in the following pages.

A portfolio manager believes that the mean returns from a mutual fund are zero. He collects the returns data over the past 100 days and calculate the sample mean returns and the sample standard deviation as follow:

x-bar = 0.2%

s = 0.50%

Based on what we have learned till now, we can state the null and alternative hypothesis as follows:

H0: m = 0 versus HA: m ≠ 0

This is a two-tailed test. We will reject the null hypothesis if the mean is not equal to 0.