One Sample Variance Test (Chi-square)

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The Chi-Square Test for One Variance is a statistical test used to compare the variance of a sample to a known population variance. It is used to test a hypothesis about the population variance and is based on the assumption that the sample is drawn from a normally distributed population.

Steps in the Chi-Square Test for One Variance:

  1. Specify the null and alternative hypotheses. The null hypothesis is usually that the population variance is equal to a specific value, while the alternative hypothesis is that the population variance is not equal to that value.
  2. Select a sample from the population and calculate the sample variance and size.
  3. Calculate the test statistic, which is the sample variance divided by the known population variance.
  4. Determine the critical value of the test statistic based on the significance level (alpha) of the test and the degrees of freedom. The degrees of freedom are calculated as the sample size minus 1.
  5. Compare the calculated test statistic to the critical value to determine whether to reject or fail to reject the null hypothesis. If the calculated test statistic exceeds the critical value, the null hypothesis is rejected, and the alternative hypothesis is accepted.

Conditions for the Chi-Square Test for One Variance:

To conduct a valid chi-square test for one variance, the following conditions must be met:

  1. The sample must be drawn randomly from the population.
  2. Each observation in the sample must be independent of the others.
  3. The population distribution must approximate a normal distribution.

Typical Null and Alternate Hypotheses in the Chi-Square Test for One Variance:

The null hypothesis in a chi-square test for one variance is that the sample variance equals the known population variance. This can be expressed as:

H0: \(σ^2 = σ_0^2\)

Where \(σ^2\) is the sample variance and \(σ_0^2\) is the known population variance.

The alternate hypothesis is the opposite of the null hypothesis and is that the sample variance is not equal to the known population variance. This can be expressed as:

Ha: \(\sigma^2 \neq \sigma_0^2\)

Calculating Test Statistic:

The chi-square statistic in a chi-square test for one variance is calculated as the sample variance divided by the known population variance. It is used to determine whether the difference between the sample and population variance is statistically significant.

The formula for calculating the chi-square statistic is as follows:

$$\LARGE{\chi^2 = \frac{(n-1)s^2}{\sigma^2}}$$

Where \(n\) is the sample size, \(s^2\) is the sample variance, and \(\sigma^2\) is the known population variance.

Calculating Critical Values:

The critical values for the chi-square statistic in a chi-square test for one variance depend on the degrees of freedom and the significance level of the test.

The degrees of freedom are calculated as the sample size minus 1.

Using these two values (significance level and degrees of freedom), you can find out the value of the critical chi-square statistic using a Chi-square table.

In addition, you can use statistical software to find out the critical value. In Excel, you can use =CHISQ.INV.RT(probability, deg_freedom) for right tail values and =CHISQ.INV(probability, deg_freedom) for left tail values.


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