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Test: steel rod diameters on two machines

Short answer

Pooled t-test (equal variances): Sp = 0.6137, t = 0.230, df = 30. At α = 0.05, critical value = ±2.042; |t| < 2.042, fail to reject H0. There is no evidence that the two machines produce rods with different mean diameters.

The diameter of steel rods manufactured on two different extrusion machines is being investigated. Two random samples of sizes n1 = 15 and n2 = 17 are selected, and the sample means and sample variances are x̄1 = 8.73, s1² = 0.35, x̄2 = 8.68, and s2² = 0.40, respectively. Assume that σ1² = σ2² and that the data are drawn from a normal distribution. Is there evidence to support the claim that the two machines produce rods with different mean diameters? Use α = 0.05 in arriving at this conclusion.

Variances are assumed equal and populations normal, with small samples, so the pooled t-test is used.

  1. H0: μ₁ = μ₂ (both machines give the same mean diameter) H1: μ₁ ≠ μ₂ (two-tailed)
  2. Level of significance: α = 0.05
  3. Test statistic (small samples, equal variances assumed, pooled t-test):
Sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2)
    = [14×0.3500 + 16×0.4000] / 30 = 0.3767
Sp  = 0.6137
t   = (x̄₁ − x̄₂) / [Sp √(1/n₁ + 1/n₂)]
    = (8.73 − 8.68) / [0.6137 × √(1/15 + 1/17)]
    = 0.0500 / 0.2174 = 0.230
  1. Critical value: t at α = 0.05, df = n₁+n₂−2 = 30: ±2.042. Reject H0 if |t| > 2.042.
  2. Decision: the calculated value falls in the acceptance region (p-value = 0.8197), so fail to reject H0.

Conclusion: there is no evidence that the two machines produce rods of different mean diameter.

✓ Answer checked by Claude Sonnet 5.5

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