Hypothesis Testing for Variance

Key Concepts

• Hypothesis Testing for Variance: Combining concepts of confidence intervals and hypothesis tests to analyze variance.
• Null Hypothesis: Typically states that the variance (or standard deviation) is equal to a specific value.
• Types of Tests: Can perform all three types (two-tailed, upper-tailed, lower-tailed) using variance.
• Critical Value vs. Test Statistic: Compare critical value from tables with a calculated test statistic.
• Test Statistic Formula: Derived using variance considerations and Chi-Square distribution.
• Chi-Square Distribution: Used when dealing with variance instead of Z or T-distributions.

Example

• Scenario: A car battery manufacturer claims the battery life is normally distributed with a standard deviation of 0.9 years. A sample shows a standard deviation of 1.2 years.
• Hypothesis: Checking if sample suggests the population standard deviation is greater than 0.9 years.
• Level of Significance: 0.05

Steps in Hypothesis Testing

1. Determine Population Parameter: Testing for variance (sigma squared) instead of standard deviation (sigma).
2. Select Probability Table: Use Chi-Square table because it involves variance.
3. Set Hypotheses:
   • Null Hypothesis (H0): ( \sigma^2 = 0.81 ) (since ( \sigma = 0.9 ), thus ( \sigma^2 = 0.81 )).
   • Alternative Hypothesis (H1): ( \sigma^2 > 0.81 ).
4. Set Up Test:
   • Calculate degrees of freedom: ( n - 1 = 9 ).
   • Determine reject region using Chi-Square table with ( \alpha = 0.05 ).
   • Critical Value: 16.919.
5. Calculate Test Statistic:
   • Formula: ( \chi^2 = \frac{(n-1) \times s^2}{\sigma^2} ).
   • Compute: ( \chi^2 = \frac{9 \times 1.2^2}{0.81} = 16.0 ).
6. Decision Rule:
   • Compare test statistic (16.0) to critical value (16.919).
   • Conclusion: 16.0 is in the "do not reject" region.
   • Conclusion Statement: Insufficient evidence to support that the variance (standard deviation > 0.9) is greater than claimed.

P-Value Method

• Calculate P-Value:
  • Based on test statistic (16.0) and degrees of freedom (9).
  • Estimated ( p )-value between 0.05 and 0.1, closer to 0.07.
• Comparison:
  • ( p )-value > ( \alpha ) (0.05), so do not reject the null hypothesis.
  • Same conclusion as with critical value method.

Conclusion

• Using either test statistic or ( p )-value methods will yield consistent conclusions in hypothesis testing for variance.
• Understanding Chi-Square distribution is crucial in variance testing.