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Test: soft drink filling machines

Short answer

Two-sample z-test gives z = (2.04−2.07)/√(0.010²/30+0.015²/30) = −9.12. Critical value ±2.576 at α=0.01; |z|>2.576, so reject H0. The machines do not fill the same mean volume; machine B fills more (2.07 L vs 2.04 L).

A soft drink is being bottled using two different filling machines. The standard deviation of the process for machine A and B was 0.010 and 0.015 L respectively. 30 bottles were randomly sampled from each machine and the means were 2.04 and 2.07 L respectively. Can one conclude that both machines are filling the same volume of soft-drink? Test the hypothesis at α = 0.01 level of significance.

The population standard deviations are known (0.010 L and 0.015 L), so a two-sample z-test is used.

  1. H0: μ_A = μ_B (both machines fill the same mean volume) H1: μ_A ≠ μ_B (two-tailed)
  2. Level of significance: α = 0.01
  3. Test statistic:
z = (x̄_A − x̄_B) / √(σ_A²/n_A + σ_B²/n_B)
SE = √(0.010²/30 + 0.015²/30) = √(0.00000333 + 0.00000750) = 0.00329
z  = (2.04 − 2.07)/0.00329 = -9.115
  1. Critical value: z(0.01, two-tail) = ±2.576. Reject H0 if |z| > 2.576.
  2. Decision: |z| = 9.115 > 2.576 (p-value < 0.0001), so reject H0.

Conclusion: the two machines do not fill the same mean volume at the 1% level; machine B fills more (2.07 L against 2.04 L).

✓ Answer checked by Claude Sonnet 5.5

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