Final answer:
To assess the need for recalibration of cereal box filling equipment, a hypothesis test comparing the observed standard deviation to the acceptable limit is suggested.
Step-by-step explanation:
To determine if recalibration of the cereal box filling equipment is necessary, we consider that the acceptable standard deviation for the weight should be 0.5 ounces or less. However, an observed standard deviation of 0.54 ounces in a sample of 84 boxes suggests that recalibration may be needed. To make this decision with statistical rigor, a hypothesis test such as a one-sample z-test could be employed, comparing the sample standard deviation to the process standard deviation.
Similarly, for assessing if a batch of apples complies with Class A weight tolerance, a hypothesis test at both the 5% and 1% significance levels could be conducted. This would involve checking if the weights of the apples in the provided batch are within 5% plus or minus of the mean weight of 150 grams. Statistical tests could include t-tests, given the relatively small sample size and assuming normal distribution of the apple weights.
For problems involving the determining of whether it might be worth driving back or managing without bags, such decisions typically involve a cost-benefit analysis which may extend beyond a strict mathematical framework, possibly touching on aspects of environmental science or economics.