Final answer:
Decreasing the sample size increases the producer's risk, also known as Type I error, in acceptance sampling by making the confidence interval wider and increasing variability, both of which make incorrectly rejecting a good lot more likely.
Step-by-step explanation:
Decreasing the sample size influences the producer's risk in acceptance sampling by potentially increasing it. Producer's risk, also known as Type I error, is the risk that a good quality lot will be rejected based on the sample results. A larger sample size tends to result in a more accurate estimate of the population parameter, which in turn increases the likelihood of making a correct decision about the lot's quality.
When the sample size decreases, the confidence interval becomes wider due to an increase in the error bound. This means there's less certainty about the true value of the population parameter, making it more likely to incorrectly reject a good lot. Furthermore, as the standard deviation of the sampling distribution of means increases with a smaller sample size, the variability among sample means is greater, which again increases the producer's risk.
In scenarios where the minimum sample size increases due to a decreased allowable error bound while maintaining the same level of confidence, the firm would need to survey more people. For example, to be 96 percent confident within one hour, the sample size must increase to achieve the desired power and to ensure that the risk of making a Type I error (producer's risk) remains acceptable.