Lecture Notes: Confidence Intervals for Variance

Overview

• Focus on confidence intervals specifically for variance.
• Discuss hypothesis testing for variance alongside confidence intervals.
• Introduction to Chi-Square distribution and how it relates to variance.

Key Concepts

Chi-Square Distribution

• Looks somewhat normal but is skewed.
• Used for confidence intervals and hypothesis testing for variance.
• Involves "1 minus Alpha" in the center with "Alpha over 2" on each edge.

Estimators and Parameters

• Estimator: ( S^2 ) (sample variance).
• Parameter being estimated: ( \sigma^2 ) (population variance).
• Related to previous discussions on confidence intervals for mean where the estimator was ( \bar{x} ) and the parameter was ( \mu ).

Chi-Square Table

• Similar to Z and T tables used for standard normal and T distributions.
• Chi-Square table reads probabilities in a different direction.
• Important to follow equation direction to avoid mistakes.

Example Problem

• Task: Find a 95% confidence interval for variance of weights of grass seed packages, assuming a normal population.

1. Formula:

   • [ \frac{n-1 \cdot S^2}{\chi^2_{\alpha/2}} < \sigma^2 < \frac{n-1 \cdot S^2}{\chi^2_{1-\alpha/2}} ]

2. Data: 10 samples given.

3. Compute Sample Variance ((S^2)):

   • [ S^2 = \frac{n \sum x_i^2 - (\sum x_i)^2}{n(n-1)} ]
   • Substitution gives: ( S^2 = 0.286 )

4. Determine Chi-Square Values:

   • Degrees of freedom ( n-1 = 9 )
   • ( \chi^2_{\alpha/2} ) for 0.025 with 9 df = 19.23
   • ( \chi^2_{1-\alpha/2} ) for 0.975 with 9 df = 2.7

5. Calculate Confidence Interval:

   • [ 9 \cdot 0.286 / 19.23 < \sigma^2 < 9 \cdot 0.286 / 2.7 ]
   • Final result: ( 0.135 < \sigma^2 < 0.953 ) (in decagrams)

6. Conclusion:

   • 95% confidence that the true population variance is between 0.135 and 0.953 decagrams.

Conclusion

• Confidence intervals for variance rely heavily on Chi-Square distributions.
• Remember to include units in final answers.
• Understanding the direction and values in Chi-Square tables is critical for accurate calculations.

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Note: Ensure familiarity with the Chi-Square distribution and table usage for accurate interpretation of variance in statistical analysis.