Test of a Single Variance

Definition

A test of a single variance is a statistical hypothesis test used to determine whether the variance of a population is equal to a specified value. It is commonly used to assess the variability or spread of a dataset and make inferences about the underlying population distribution.

5 Must Know Facts For Your Next Test

1. The test of a single variance is used to assess the variability or spread of a dataset and make inferences about the underlying population distribution.
2. The test statistic used in a test of a single variance is calculated as $(n-1)s^2/\sigma^2$, where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma^2$ is the hypothesized population variance.
3. The test statistic follows a chi-square distribution with $n-1$ degrees of freedom under the null hypothesis.
4. The null hypothesis is rejected if the calculated test statistic falls in the critical region, which is determined by the chosen significance level and the degrees of freedom.
5. The test of a single variance can be used to determine if the population variance is significantly different from a hypothesized value, which is important in quality control, process monitoring, and other applications.

Review Questions

• Explain the purpose and underlying assumptions of the test of a single variance.
  • The purpose of the test of a single variance is to determine whether the variance of a population is equal to a specified value. The underlying assumptions are that the data follows a normal distribution and the sample is randomly selected from the population. The test statistic used follows a chi-square distribution, and the null hypothesis is that the population variance is equal to the specified value, while the alternative hypothesis is that the population variance is not equal to the specified value.
• Describe the steps involved in conducting a test of a single variance.
  • The steps involved in conducting a test of a single variance are: 1) Specify the null and alternative hypotheses, 2) Calculate the test statistic using the formula $(n-1)s^2/\sigma^2$, where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma^2$ is the hypothesized population variance, 3) Determine the critical value from the chi-square distribution based on the chosen significance level and the degrees of freedom, which is $n-1$, 4) Compare the calculated test statistic to the critical value and make a decision to either reject or fail to reject the null hypothesis.
• Discuss the practical applications and implications of the test of a single variance in various fields.
  • The test of a single variance has numerous practical applications across various fields, such as quality control, process monitoring, and experimental design. In quality control, it can be used to assess the variability of a manufacturing process and determine if it meets specified standards. In process monitoring, the test can be used to detect changes in the variability of a process over time, which can indicate potential issues. In experimental design, the test of a single variance can be used to determine the appropriate sample size and ensure that the experiment has sufficient statistical power to detect meaningful differences in the population variance. The implications of the test results can inform decision-making, process improvements, and the design of future studies.

Related terms

Null Hypothesis: The null hypothesis in a test of a single variance is that the population variance is equal to a specified value, typically denoted as $\sigma^2$.

Alternative Hypothesis: The alternative hypothesis in a test of a single variance is that the population variance is not equal to the specified value, which can be either greater than or less than the null hypothesis value.

Chi-Square Distribution: The test statistic used in a test of a single variance follows a chi-square distribution, which is a continuous probability distribution used to make inferences about the variance of a population.