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Test: heat capacity of coal from two mines

Short answer

Pooled t-test assuming equal variances: sp²=13066.67, t=4.19, df=9, critical t=±3.250. |t| > 3.250, reject H0; difference in mean heat capacities is significant at 1% level.

The following random samples are measurement of the heat producing capacity (in millions of calories per ton) of specimens of coal from two mines:

Mine 182608130835080708340-
Mine 2795078907900814079207840
Use the 0.01 level of significance to test whether the difference between means of these two samples is significant.

Samples are small and independent, so a pooled t-test is used (populations assumed normal with equal variances).

Sample statistics:

n₁ = 5, Σx₁ = 41150, x̄₁ = 8230.0000, s₁² = 15750.0000
n₂ = 6, Σx₂ = 47640, x̄₂ = 7940.0000, s₂² = 10920.0000
  1. H0: μ₁ = μ₂ (mean heat capacities are equal) H1: μ₁ ≠ μ₂ (two-tailed)
  2. Level of significance: α = 0.01
  3. Test statistic (small samples, equal variances assumed, pooled t-test):
Sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2)
    = [4×15750.0000 + 5×10920.0000] / 9 = 13066.6667
Sp  = 114.3095
t   = (x̄₁ − x̄₂) / [Sp √(1/n₁ + 1/n₂)]
    = (8230.0 − 7940.0) / [114.3095 × √(1/5 + 1/6)]
    = 290.0000 / 69.2179 = 4.190
  1. Critical value: t at α = 0.01, df = n₁+n₂−2 = 9: ±3.250. Reject H0 if |t| > 3.250.
  2. Decision: the calculated value falls in the rejection region (p-value = 0.0023), so reject H0.

Conclusion: the difference between the mean heat-producing capacities of the two mines is significant at the 1% level.

✓ Answer checked by Claude Sonnet 5.5

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