Question

A research study reports the following data on the strength measurement of two independent batches of ceramic tiles (×109 MPa).

Batch 1: 1.75 2.12 2.05 1.97
Batch 2: 1.77 1.59 1.70 1.69
Obtain a 95% confidence interval for the ratio of variances.

Answer 1

The 95% confidence interval for the ratio of variances is (0.0293, 0.7125).

For Batch 1:

Mean (1) = (1.75 + 2.12 + 2.05 + 1.97) / 4

= 1.975

Variance (s₁²) = [(1.75 - 1.975)² + (2.12 - 1.975)² + (2.05 - 1.975)² + (1.97 - 1.975)²] / (4 - 1)

s₁² = 0.0392

For Batch 2:

Mean (2) = (1.77 + 1.59 + 1.70 + 1.69) / 4

= 1.69

Variance (s₂²) = [(1.77 - 1.69)² + (1.59 - 1.69)² + (1.70 - 1.69)² + (1.69 - 1.69)²] / (4 - 1)

s₂² = 0.0031

F_statistic = s₁² / s₂² = 0.0392 / 0.0031

= 12.65

Degrees of freedom (df_₁): n₁ - 1 = 4 - 1 = 3

Degrees of freedom (df_₂): n₂ - 1 = 4 - 1 = 3

Confidence level: 95%

We then use an F_distribution table or calculator, we find the upper and lower critical F_values:

F_upper (0.95, 3, 3) = 9.28

F_lower (0.05, 3, 3) = 0.39

Lower bound: F_lower * s₂² / s₁² = 0.39 * 0.0031 / 0.0392 = 0.0293

Upper bound: F_upper * s₂² / s₁² = 9.28 * 0.0031 / 0.0392 = 0.7125