Welcome back to the AI Bayeslab Statistics series.

We demonstrated the chi-squared application with a simple quality control example. Now, let's examine a real-world case using the F-distribution, which is fundamentally used for statistical tests involving variance.

As previously mentioned, the F-test determines variance homogeneity when population variances are unknown. This preliminary step is crucial because we need to ensure the two sample groups share similar characteristics before applying any experimental intervention.

Consider the following scenario: we have two sample groups, where only one experimental condition is modified (e.g., a product upgrade). To attribute any observed differences to this specific change, we must first verify that the groups were initially comparable in all relevant aspects. One group receives the intervention (the experimental group), while the other remains unchanged (the control group). After the intervention, if we observe significant differences between the groups that weren't present initially, we can reasonably conclude these differences resulted from our designed change. This is precisely why we need to test for variance homogeneity between sample groups before analysis.

2. Pharmaceutical Drug Comparison

Two blood pressure medications (Drug A and Drug B) are tested on 50 patients each. Drug A shows a variance in effectiveness (s₁² = 25), while Drug B shows s₂² = 36. Is there a statistically significant difference in the consistency of their effects? (F-test for variance equality: H₀: σ₁² = σ₂² vs. H₁: σ₁² ≠ σ₂²)

3. Production Line Consistency Check

A factory has two production lines, A and B, that make the same type of battery. A sample of 30 batteries from Line A shows a lifespan variance of 8 hours², while a sample of 25 from Line B shows a variance of 12 hours². Do the two production lines have the same variability in battery lifespan? (F-test for homogeneity of variances)

Note: Our actual data is based on our original data file, and the calculated actual variance slightly differs from the variance in the examples above.

The two examples above come from our previous article "Real-Life with Inferences About Population Variance: Chi-Squared and F-Dis." The last two cases mentioned pertain to the F-distribution. Today, we will primarily focus on the issues in these two cases. Generally, testing whether there is a significant difference between the variances of two normal populations is referred to as a "test for homogeneity of variances," and the F-test can be employed for this purpose.

• If X₁ ~X²₍𝑑𝑓₁₎ and X₂~X²₍𝑑𝑓₂₎, and ( X₁ ) and ( X₂ ) are independent, then the random variable F(x) can be expressed as：

  

F = \frac{X_1 / \text{df}_1}{X_2 / \text{df}_2} \sim F_{\text{df}_1, \text{df}_2}

Now, we randomly draw two independent samples from two populations (X₁) and (X₂) and calculate the corresponding (S²) and (S₂²). If the variances of (X₁) and (X₂) are equal, i.e., (σ₁² = σ₂²), then the ratio of (s₁²) and (s₂²) follows an F-distribution.

F = \frac{s_1^2}{s_2^2}

F-Statistic Formula Explained

The F-statistic is a ratio of two sample variances, used primarily in hypothesis testing to compare whether two populations have the same variance.

Formula:

F = \frac{\text{Larger Sample Variance}}{\text{Smaller Sample Variance}} = \frac{\max(s_1^2, s_2^2)}{\min(s_1^2, s_2^2)}

Key Rules:

1. Always Ensure F ≥ 1

   1. By convention, we place the larger sample variance in the numerator and the smaller one in the denominator.

   2. This guarantees F ≥ 1, making interpretation and critical value comparison easier.

   3. Example:

      • If s₁²=21.7932 and s₂²= 27.5192, then: F=\frac{21.793}{27.5192}=0.7919

   4. If s₂² = 21.7932 and s₁² = 27.5192 , then: F=\frac{21.793}{27.5192}=0.7919

   
   

2. Degrees of Freedom (df)

   1. The numerator degrees of freedom df_1, correspond to the sample size of the group with the larger variance:df_1 = n_{\text{larger variance group}} - 1

   
   

3. The denominator degrees of freedom df_2, correspond to the sample size of the group with the smaller variance:df_2 = n_{\text{smaller variance group}} - 1

   
   

4. Interpretation:

   1. If F ≈ 1, the variances are likely equal, accept H₀; in other words, we perceive the two populations as homogeneous (σ₁² = σ₂²).

   2. If F is significantly greater than 1, the variances differ, possibly leading to the rejection of H₀; in other words, we perceive the two populations as heterogeneous (σ₁² ≠ σ₂²).

Example 1: Pharmaceutical Drug Comparison‘

The F value calculated from our statistics is 0.7919, which falls within the acceptance region (05675, 1.7622). Therefore, we conclude that there is no statistically significant difference in the consistency of their effects.

Example 2: Production Line Consistency Check

Similarly, in Case 2, the calculated F value for the production line is 0.5903, which also falls within the acceptance region (0.464, 2.217). Here, we still conclude that the battery lifespans produced by the two production lines have similar quality differences.

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