Interpret the key results for 1 Variance

Complete the following steps to interpret a 1 variance test. Key output includes the estimate of the standard deviation or variance, the confidence interval, and the p-value.

Step 1: Determine a confidence interval for the population standard deviation or variance

First, consider the sample variance or the sample standard deviation, and then examine the confidence interval.

The variance and the standard deviation of your sample data provide an estimate of the population variance and the population standard deviation. Because the standard deviation and the variance are based on sample data and not on the entire population, it is unlikely that the sample standard deviation and the sample variance equal the population standard deviation and the population variance. To better estimate the population standard deviation and the population variance, use the confidence interval.

The confidence interval provides a range of likely values for the population standard deviation or the population variance. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population standard deviation or the population variance. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size. For more information, go to Ways to get a more precise confidence interval.

Note

When you enter a column of data, Minitab only calculates a confidence interval for the standard deviation.

Minitab displays two confidence intervals. Usually, you should use the Bonett method. Use the chi-square method only when you are certain that the data follow a normal distribution. Any small deviation from normality can greatly affect the chi-square method results. The chi-square method is provided for teaching purposes where you can theoretically assume normality without any practical consequences.
Note

Minitab cannot calculate the Bonett method with summarized data.

Descriptive Statistics 95% CI for σ 95% CI for σ using N StDev Variance using Bonett Chi-Square 50 0.871 0.759 (0.704, 1.121) (0.728, 1.085)
Key Results: StDev, Variance, CI for StDev, CI for Variance

In these results, the estimate of the population standard deviation for the length of beams is 0.871, and the estimate of the population variance is 0.759. Because the data did not pass a normality test, use the Bonett method. You can be 95% confident that the population standard deviation is between 0.704 and 1.121.

Step 2: Determine whether the difference between the population variance or standard deviation and the hypothesized variance or standard deviation is statistically significant

To determine whether the difference between the population variance or the population standard deviation and the hypothesized value is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
P-value ≤ α: The difference between the variances or standard deviations is statistically significant (Reject H0)
If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis. You can conclude that the difference between the population variance or standard deviation the hypothesized variance or standard deviation is statistically significant. Use your specialized knowledge to determine whether the difference is practically significant. For more information, go to Statistical and practical significance.
P-value > α: The difference between the variances or standard deviations is not statistically significant (Fail to reject H0)
If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis. You do not have enough evidence to conclude that the difference between the population variance or standard deviation and the hypothesized variance or standard deviation is statistically significant. You should make sure that your test has enough power to detect a difference that is practically significant. For more information, go to Power and Sample Size for 1 Variance.
Minitab displays two p-values. Usually, you should use the Bonett method. Use the chi-square method only when you are certain that the data follow a normal distribution. Any small deviation from normality can greatly affect the chi-square results. The chi-square method is provided for teaching purposes where you can theoretically assume normality without any practical consequences.
Note

When you have summarized data, Minitab cannot calculate a p-value for the Bonett method.

Test Null hypothesis H₀: σ = 1 Alternative hypothesis H₁: σ ≠ 1
Test Method Statistic DF P-Value Bonett — — 0.275 Chi-Square 37.17 49 0.215
Key Result: P-Value

In these results, the null hypothesis states that standard deviation of beam lengths equals 1. Because the data did not pass a normality test, use the p-value for the Bonett method. Because the p-value of 0.275 is greater than the significance level of 0.05, you fail to reject the null hypothesis and cannot conclude that the standard deviation is different from 1.

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