Rice Consignment: One-Tailed or Two-Tailed Test Under a Contract Clause
A cooperative retail chain buys 500 g packets of rice from a mill. A random sample of 25 packets from the latest consignment has a mean weight of 492 g and a standard deviation of 20 g. The purchase contract says a consignment may be rejected if the mean packet weight is significantly below 500 g, and the chain has always tested at the 5 per cent level. The mill's quality officer argues that the test must be two-tailed because the clause says the mean may 'differ' from 500 g, and that in any case a single sample of 25 proves little. The purchase manager disagrees and wants a written decision. Internal costing shows that wrongly rejecting a good consignment costs about Rs 40,000 in emergency buying, while accepting an underweight consignment costs about Rs 60,000 in rework, refunds and reputation. The category head has asked for the test result, the correct framing of the hypothesis, and a defensible decision rule.
1. Carry out the test in the form the mill's officer proposes (two-tailed, 5 per cent) and state the decision with the critical value used.
Read the worked answer
Hypotheses as the mill frames them: H0: mean = 500 g against H1: mean not equal to 500 g.
Standard error = s / square root of n = 20 / square root of 25 = 20/5 = 4 g.
t = (492 - 500)/4 = -2.00 with 24 degrees of freedom.
At the 5 per cent level for a two-tailed test, the critical value is t(0.025, 24) = 2.064. Since |-2.00| = 2.00 < 2.064, we do not reject H0. The two-tailed p-value is about 0.057, just above 5 per cent.
So under the mill's framing the consignment escapes rejection, but only by a very narrow margin. That narrowness matters for the decision, because a slightly larger sample or a slightly smaller standard deviation would reverse the result. The officer's framing therefore does not prove the consignment is acceptable; it only shows that the evidence is not strong enough to call the shortfall two-sided.
2. Re-test using the framing implied by the contract, and explain which framing is correct and why.
Read the worked answer
The contract only penalises a mean significantly below 500 g, so the correct test is one-tailed: H0: mean is at least 500 g against H1: mean < 500 g.
The test statistic is unchanged: t = -2.00 with 24 degrees of freedom. The one-tailed critical value at 5 per cent is -1.711. Since -2.00 < -1.711, we reject H0 and conclude that the mean packet weight is significantly below 500 g. The one-tailed p-value is about 0.028, roughly half the two-tailed value, as expected.
The one-tailed framing is correct because the contract is directional: an overweight consignment is not a breach, so it should not contribute to the rejection region. The mill's officer has made the classic error of choosing the two-tailed test because it is harder to reject, not because the business question is two-sided. Two cautions should be recorded: the direction must be fixed by the contract before the data are seen, not chosen after, and the test assumes the packet weights are roughly normal, which is reasonable for a machine-filled product.
3. Suppose the sample standard deviation had been 12 g rather than 20 g. What would change, and what does this show about statistical versus practical significance?
Read the worked answer
With s = 12 g, the standard error becomes 12/5 = 2.4 g, and t = (492 - 500)/2.4 = -3.33 with 24 degrees of freedom. This exceeds both the one-tailed critical value of 1.711 and the two-tailed value of 2.064, so H0 is rejected under either framing, with a two-tailed p-value well below 0.01.
The 8 g shortfall is the same in both samples; only its variability changed. The lesson is that statistical significance measures the shortfall relative to sampling noise, not the size of the shortfall: a consistent 8 g gap is strong evidence of a real gap, while a noisy 8 g gap could easily be chance. For the purchase decision, combine three things: the directional test result, the cost asymmetry (Rs 40,000 for wrongful rejection versus Rs 60,000 for wrongful acceptance), and the practical size of the shortfall. With the original data the evidence is just strong enough to reject at 5 per cent, so the pragmatic move is to withhold payment or demand a price adjustment rather than reject the whole consignment outright, and to raise the sample size for the next lot so that the decision does not hinge on a margin of 0.06 in the t-ratio.