Question

Which ONE or MORE among the following options is/are TRUE for the ideal operating characteristics (OC) curve given in the figure?

Note: \( p_0 \) represents the acceptable quality level (AQL) and \( p_1 \) represents the lot tolerance percent defective (LTPD).

• A. Producer's risk is 100%.
• B. Consumer's risk is 100%.
• C. Producer's risk is zero.
• D. Consumer's risk is zero.

Hint:

On the ideal OC curve, acceptance probability is 1 below p0 and 0 above p1, leaving no room for wrongly rejecting good lots or wrongly accepting bad ones.

Answer 1

The Correct Option is C, D

Step 1: Recall What an OC Curve Tells Us:
An OC curve plots the probability that a sampling plan accepts a lot, $P_a$, against the true fraction defective $p$ in that lot.
A real sampling plan gives an S-shaped curve because a sample can never perfectly represent the whole lot, but the 'ideal' curve in this figure represents a hypothetical plan with zero sampling error, a straight vertical drop instead of a smooth S-curve.

Step 2: Working with Probability of Rejection Instead:
Since $P_a + P_{reject} = 1$ at every value of $p$, we can find the rejection probability just as easily as the acceptance probability from the graph.
Below $p_0$: the graph shows $P_a = 1$, so $P_{reject} = 1 - 1 = 0$. Since producer's risk is the chance of a good lot ($p \le p_0$) being rejected, and $P_{reject} = 0$ throughout this whole region, the producer's risk is 0, not 100 percent. This confirms (C) is correct and rules out (A).
Above $p_1$: the graph shows $P_a = 0$ directly. Since consumer's risk is the chance of a bad lot ($p \ge p_1$) being accepted, and $P_a = 0$ throughout this whole region, the consumer's risk is 0, not 100 percent. This confirms (D) is correct and rules out (B).

Step 3: Why This Makes Sense:
The word 'ideal' here means the sampling plan behaves like a perfect 100 percent inspection: every lot at or below the AQL passes, every lot at or above the LTPD fails, with nothing in between except the theoretical jump from $p_0$ to $p_1$.
Real plans cannot achieve this because a sample never tells you the lot's true defect rate with certainty, which is exactly why real OC curves are smooth S-curves with nonzero risks on both sides.

Final Answer:
Working from $P_{reject} = 1 - P_a$ at both ends of the curve confirms both risks are zero for the ideal case. \[ \boxed{\alpha = 0, \ \beta = 0 \implies \text{options C and D}} \]