Introduction




You might have learned about testing the population mean/s. In this article, you will learn how to perform the hypothesis test for population standard deviation. Consider the following scenarios.

Studying the volatility in the rates of returns for a stock A, testing whether the standard deviation of the volume of the water tanks is different from 1 cm3, testing the variance of weights of people in a gym, etc. In all these situations, either the population variance or the standard deviation are parameter/s of interest.

To test the claim about population variance or population standard deviation, we collect the sampled data from population/s. Then we make the inferences and conclusions based on these samples. There are some assumptions that should be satisfied to carry out variance testing.


Assumptions for chi-square test for variation

  1. The data should be a simple random sample.
  2. The observations in the data should be independent.
  3. The population should follow the normal distribution.

In addition, if the sample size is n < 0.05*N then this assures the observations are independent.

If the data follows all these conditions, then one can use the chi-square variance test. Using the chi-square distribution, you can test the claim about population variance or population standard deviation for a single population. The chi-square test is not applicable for comparing two population variances/standard deviations.


Steps

Let us see what the steps are involved in the chi-square test for variation one by one.


Step 1: Fix the null and alternative hypotheses

The null and alternative hypotheses for proportion testing are as follows.



Step 2: Chose the correct formula to calculate the test statistic

Use the following formula to find the test statistic for the chi-square test for variation. The same formula is used while testing population variance and population standard deviation.

Here s² is the sample variance, σ02 is the standard deviation and n is the sample size. The test statistic in chi-square tests follows the chi-square distribution with (n-1) degrees of freedom.


Step 3: Fix the significance level (α)

Most of the time the value of α is given in the question. If you do not know the value of α then just take α=0.05.


Step 4: Find the critical value/s or p-value

The critical values or p-value are essential in making the decision, whether to reject or fail to reject the null hypothesis. You can use either the statistical tables or technology like excel or TI84 calculator or R-Studio, etc. to find critical values as well as p-value.


Step 5: Make the decisions based on critical value/ p-value

Based on critical values:

Using critical values, you can decide the rejection region. If the value of test statistic found in step 2 lies in the rejection region, then reject the null hypothesis at α level of significance, otherwise fail to reject the null hypothesis.

Based on p-value:

If the p-value ≤ α then reject the null hypothesis otherwise fail to reject the null hypothesis.


Step 6: Conclusion based on the decision

If you reject the null hypothesis then write that, “There is sufficient evidence to support the claim in alternative hypothesis”. This is exactly the same as writing, “There is sufficient evidence to reject the claim in null hypothesis”.

If you fail to reject the null hypothesis then write that, “There is insufficient evidence to support the claim in alternative hypothesis”. This is exactly the same as writing, “There is insufficient evidence to reject the claim in null hypothesis”.

Write the conclusion based on the test that you are using.