Hypothesis Testing

Hypothesis Testing (Table of Contents)

• Formula
• Examples

What is the Hypothesis Testing in Statistics?

Hypothesis Testing is a statistical method done in order to test a particular theory and has two parts one is known as the null hypothesis and the other is known as the alternative hypothesis. The null hypothesis is the one that the researcher tries to reject. It is difficult to prove the alternate hypothesis, so if the null hypothesis is rejected the remaining alternate hypothesis gets accepted.

For example, if we believe that the returns from the NASDAQ stock index are not zero.  Then the null hypothesis, in this case, is that the return from the NASDAQ index is zero.

Hypothesis Testing Formula

As we already discussed the two important parts here is the null hypothesis and the alternate hypothesis. The formula to measure the null hypothesis and the alternate hypothesis involves defining the null hypothesis and the alternate hypothesis.

H0:µ0=0

Ha: µ0≠0

Where

• H0 = null hypothesis
• Ha = alternate hypothesis

We will also need to calculate the test statistic in the case of hypothesis testing in order to be able to reject the hypothesis testing.

The formula for the test statistic is represented as follows,

Detailed Explanation

It is a statistical method done to test a particular hypothesis. It has two parts one is known as the null hypothesis and the other is known as the alternative hypothesis. The null hypothesis is the one that the researcher tries to reject. It is difficult to prove the alternate hypothesis, so if the null hypothesis is rejected the remaining alternate hypothesis gets accepted. It is tested at a different level of significance will the help of calculating the test statistics.

Examples of Hypothesis Testing Formula

Let’s see some examples to understand it better.

Example #1

Let us try to understand the concept of hypothesis testing with the help of an example. Suppose we want to know that the mean return from a portfolio over a 200 day period is greater than zero. The mean daily return of the sample is 0.1% and the standard deviation is 0.30%.

In this case, the null hypothesis which the researcher would like to reject is that the mean daily return for the portfolio is zero. The null hypothesis, in this case, is a two tail test. We will be able to reject the null hypothesis if the test statistic is outside the range of the level of significance.

At a 10% level of significance, the z-value for the two-tailed test will +/- 1.645. So if the test statistic is beyond this range then we will reject the hypothesis.

Based on the given information, determine the test statistic

Therefore, the calculation of test statistic will be as follows,

T= µ/(s/√n)

=0.001/(0.003/√200)

Test Statistic will be –

The test statistic is = 4.71

Since the value of the test statistic is more than +1.645 then the null hypothesis will be rejected for a 10% level of significance. Therefore the alternate hypothesis is accepted for the research that the mean value of the portfolio is greater than zero.

Example #2

Let us try to understand the concept of hypothesis testing with the help of another example. Suppose we want to know that the mean return from a mutual fund over a 365 day period is greater than zero. The mean daily return of the sample if 0.8% and the standard deviation is 0.25%.

In this case, the null hypothesis which the researcher would like to reject is that the mean daily return for the portfolio is zero. The null hypothesis, in this case, is a two tail test. We will be able to reject the null hypothesis if the test statistic is outside the range of the level of significance.

At 5% level of significance, the z-value for the two-tailed test will +/- 1.96. So if the test statistic is beyond this range then we will reject the hypothesis.

Below is the given data for calculation of test statistic

Therefore, the calculation of test statistic will be as follows,

T= µ/(s/√n)

=.008/(.025/√365)

Test Statistic will be –

Test Statistics = 61.14

Since the value of the test statistic is more than +1.96 then the null hypothesis will be rejected for a 5% level of significance. Therefore the alternate hypothesis is accepted for the research that the mean value of the portfolio is greater than zero.

Example #3

Let us try to understand the concept of hypothesis testing with the help of another example for a different level of significance. Suppose we want to know that the mean return from an options portfolio over a 50 day period is greater than zero. The mean daily return of the sample if 0.13% and the standard deviation is 0.45%.

In this case, the null hypothesis which the researcher would like to reject is that the mean daily return for the portfolio is zero. The null hypothesis, in this case, is a two tail test. We will be able to reject the null hypothesis if the test statistic is outside the range of the level of significance.

At a 1% level of significance, the z-value for the two-tailed test will +/- 2.33. So if the test statistic is beyond this range then we will reject the hypothesis.

Use the following data for the calculation of test statistic

So, the calculation of test statistic can be done as follows-

T= µ/(s/√n)

=.0013/ (.0045/√50)

Test Statistic will be –

The test statistic is = 2.04

Since the value of the test statistic is less than +2.33 then the null hypothesis cannot be rejected for a 1% level of significance. Therefore the alternate hypothesis is rejected for the research that the mean value of the portfolio is greater than zero.

Relevance and Use

It is a very important test to validate a theory. In practice it is difficult to validate a theory statistically, that’s why a researcher tries to reject the null hypothesis in order to validate the alternate hypothesis. It plays an important role in accepting or rejecting decisions in businesses.

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