Concrete Lot Acceptance With One Low Cylinder
A ready-mix supplier delivers 60 m³ of concrete for a bridge deck. The specification requires a 28-day design strength of 28 MPa. Twelve standard cylinders were cast and tested at 28 days: the mean is 31.2 MPa with a sample standard deviation of 3.8 MPa. One cylinder tested at 24.1 MPa. The contractor wants to strip forms and start the next span immediately; the owner's inspector wants to reject the whole lot and has asked you, as the independent reviewer, for a recommendation. Truck tickets and the cylinder log are consistent, sampling and curing records show no errors, and there is neither time nor budget for a second round of tests before the scheduled pour. Rejection would cost roughly $95,000 in removal and rework and delay the span by three weeks.
1. Compute a 95% one-sided lower confidence limit on the true mean strength (use t = 1.796 for 11 degrees of freedom) and state whether the mean-strength criterion is satisfied.
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Lower confidence limit on the mean
$$LCL = \bar{x} - t_{0.95,11},\frac{s}{\sqrt{n}} = 31.2 - 1.796\cdot\frac{3.8}{\sqrt{12}}$$
$$\frac{s}{\sqrt{n}} = \frac{3.8}{3.464} = 1.097 \text{ MPa}$$
$$LCL = 31.2 - 1.796(1.097) = 31.2 - 1.97 = 29.2 \text{ MPa}$$
Since 29.2 MPa > 28 MPa, the mean-strength criterion is met with a statistically defensible margin. Note that the point estimate (31.2) overstates confidence: the honest statement is "the true mean is at least 29.2 MPa with 95% confidence," which is only 1.2 MPa of margin, not 3.2 MPa.
2. Is the single 24.1 MPa cylinder by itself evidence of a deficient lot? Quantify the argument.
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Model the individual results as normal with mean 31.2 MPa and standard deviation 3.8 MPa (this understates spread, since s is the sample sd, but it is the right first-order check).
$$z = \frac{28 - 31.2}{3.8} = -0.842 \Rightarrow P(X < 28) \approx 0.20$$
Out of 12 cylinders you would expect about 12 × 0.20 ≈ 2.4 results below 28 MPa purely from natural variability, even though the lot mean comfortably exceeds the specification. Observing one such cylinder is therefore less than the expected number, not more.
The 24.1 MPa value has z = (24.1 − 31.2)/3.8 = −1.87, about a 3% tail probability per cylinder - unusual but not extraordinary in 12 draws (roughly a 30% chance of seeing at least one such value). Conclusion: the single low cylinder is not, on its own, statistical evidence of a deficient lot. It is a signal worth recording, not a rejection trigger.
3. Recommend an action, weighing the $95,000 rework cost and three-week delay against risk to the owner. State what additional evidence, if any, you would require.
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Recommendation: accept the lot for strength, with conditions. The basis is that the acceptance decision is about the mean of the production, and the 95% lower bound (29.2 MPa) clears the 28 MPa requirement. Rejecting on a single low individual result would be a decision rule with a high false-alarm rate (~2.4 expected low cylinders per 12) and would cost $95,000 plus three weeks for no measurable increase in structural safety.
Conditions I would attach:
- Document the low cylinder in the lot record and require that the affected truck batch not be used in a critical region if the pour is still segregable.
- Tighten the acceptance rule going forward - the current rule passes a lot whose true mean could be as low as 29.2 MPa, only 4% above design. Ask for a target mean of ~33–34 MPa from the supplier so the lower bound sits comfortably above 28.
- If the owner still objects, the proportionate next step is nondestructive testing or cores from the low-strength region - a few thousand dollars, not a $95,000 demolition - and only if cores fail does removal become justified.
Framing for the owner: the question is not "is any cylinder low" but "does the evidence show the in-place concrete is below design." It does not.