Testing a seed lot's germination claim and a farmer's 25-seed rule
A seed company states that a tomato lot has 92% germination. A public laboratory randomly draws 400 seeds, sows them under standard conditions, and obtains 352 germinated seedlings. The seeds are from a single lot and are treated as a random sample. The company argues that the difference is just sampling noise. A farmer who buys small packets uses a simpler rule: take 25 seeds and accept the lot only if at least 21 germinate. The farmer wants no more than a 5% chance of rejecting a lot that truly has 92% germination, and assumes the 25 seeds are independent. The laboratory reports a two-sided test at alpha = 0.05. The farmer asks whether the laboratory evidence contradicts the company claim, and whether the 25-seed rule is a reliable protection against a lot with lower germination.
1. Carry out the two-sided test of the company's 92% claim at alpha = 0.05. State the hypotheses, test statistic, P-value interpretation and conclusion.
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Let p be the true germination probability. H0: p = 0.92; Ha: p ≠ 0.92. The sample proportion is phat = 352/400 = 0.88. Under H0, SE = sqrt(0.92 × 0.08 / 400) = sqrt(0.000184) = 0.01356. z = (0.88 - 0.92)/0.01356 = -2.95. For a two-sided test, the P-value is about 0.0032. This is below 0.05, so reject H0. The data provide strong evidence that the lot's germination is below 92%. A 95% confidence interval is approximately 0.88 ± 1.96 × 0.01356, or 0.853 to 0.907, which lies below 0.92. The conclusion assumes random sampling and independent germination outcomes; if the sample was not random, the inference is weaker.
2. Assuming the true germination is 92%, compute the probability that the farmer's rule accepts the lot. Then comment on whether the rule protects the farmer well against a lot with 88% germination.
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Let X be the number germinating among 25 seeds. Under p = 0.92, X is Binomial(25, 0.92). The rule accepts if X ≥ 21. P(X ≥ 21) = sum from k = 21 to 25 of C(25,k) × 0.92^k × 0.08^(25-k). The terms are approximately: k=25: 0.124; k=24: 0.270; k=23: 0.282; k=22: 0.188; k=21: 0.090. Sum ≈ 0.955. So the farmer rejects a truly 92% lot only about 4.5% of the time, satisfying the stated 5% risk. However, the rule is weak against a lower-quality lot. If the true p = 0.88, P(X ≥ 21) is about 0.826, so the farmer still accepts a lot with 88% germination more than 80% of the time. The 25-seed sample is too small to discriminate reliably between 92% and 88%; a larger sample, or a rule requiring more than 21 germinating seeds, would give better protection.