Finding Critical Values for a 90% Confidence Interval Using the Chi-Square Distribution
In statistics, the Chi-Square distribution is frequently used to estimate population variances and test hypotheses involving categorical data. One of its important applications is in constructing confidence intervals for variances. In this blog, we will demonstrate how to find the critical values for a 90% confidence interval using the Chi-Square distribution with 13 degrees of freedom.
Problem Overview
We are tasked with finding the critical values for a 90% confidence interval with a Chi-Square distribution and 13 degrees of freedom. These critical values define the boundaries of the interval within which we can estimate the population variance or standard deviation.
Step-by-Step Calculation
We can break down the problem into clear steps:
- Determine the Level of Significance: The confidence level is 90%, so the level of significance $ (\alpha) $ is calculated as:
$ \alpha = 1 – 0.90 = 0.10 $
Since we are dealing with a two-tailed test, we split the significance level in half:
$ \frac{\alpha}{2} = 0.05 $ - Identify the Degrees of Freedom:
The degrees of freedom $ (df) $ are provided as 13. This will be used to find the critical values from the Chi-Square distribution table or using statistical software like Excel. - Find the Lower Critical Value:
The lower critical value $ (\chi^2_L) $ corresponds to the (1 – $\frac{\alpha}{2}$) percentile of the Chi-Square distribution with 13 degrees of freedom:
$ \chi^2_L = \chi^2_{(1 – \frac{\alpha}{2}, df)} = \chi^2_{(1 – 0.05, 13)} = \chi^2_{(0.95, 13)} $
Using a Chi-Square distribution table or Excel functionCHISQ.INV.RT(0.05, 13), we find:
$ \chi^2_L \approx 5.892 $ - Find the Upper Critical Value:
The upper critical value $ (\chi^2_U) $ corresponds to the $\frac{\alpha}{2}$ percentile of the Chi-Square distribution with 13 degrees of freedom:
$ \chi^2_U = \chi^2_{\frac{\alpha}{2}, df} = \chi^2_{(0.05, 13)} $
Using a Chi-Square distribution table or Excel functionCHISQ.INV(0.05, 13), we find:
$ \chi^2_U \approx 22.362 $
Final Answer
The critical values for a 90% confidence interval with 13 degrees of freedom are approximately:
- Lower Critical Value: $ \chi^2_L \approx 5.892 $
- Upper Critical Value: $ \chi^2_U \approx 22.362 $
Why Chi-Square Distribution Matters
The Chi-Square distribution is particularly useful when working with sample variances and performing goodness-of-fit tests. It’s widely applied in fields like quality control, biology, and economics, where we need to estimate population variance based on sample data.
Knowing how to find the critical values for a Chi-Square distribution allows you to confidently estimate a population parameter within a specific range, especially when dealing with variances.
Conclusion
In this blog, we walked through the steps to find the critical values for a 90% confidence interval using the Chi-Square distribution. With 13 degrees of freedom, the lower and upper critical values are approximately 5.892 and 22.362, respectively. Understanding this process is crucial for estimating population variances and performing hypothesis tests in real-world data analysis.
