- What is Hypothesis Testing
- Test Statistic, Type I and type II Errors, and Significance Level
- Decision Rule in Hypothesis Testing
- p-Value in Hypothesis Testing
- Selecting the Appropriate Test Statistic
- Hypothesis Testing with t-statistic
- Hypothesis Testing with z-statistic
- Tests Concerning Differences in Means
- Paired Comparision Tests - Mean Differences When Populations are Not Independent
- Hypothesis Tests Concerning Variances
- Chi-square Test – Test for value of a single population variance
- F-test - Test for the Differences Between Two Population Variances
- Non-parametric Tests

# What is Hypothesis Testing

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:

- Stating the hypotheses.
- Identifying the appropriate test statistic and its probability distribution.
- Specifying the significance level.
- Stating the decision rule.
- Collecting the data and calculating the test statistic.
- 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:

- H0: θ = θ0 versus HA: θ ≠ θ0
- H0: θ ≤ θ0 versus HA: θ > θ0
- 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.

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