Suppose we work for a company that makes a popcorn product containing 1.2 ounces of unpopped kernels in a microwavable bag. However, no manufacturing process is perfect, and there is variability from bag to bag, which the factory manager seeks to keep to a minimum. Bags that are underfilled can lead to consumer complaints and lawsuits, while bags that are overfilled can result in lost profits and affect the quality of the popping process. Based on previous experience, the distribution of the popcorn bags is distributed normally with a mean of 1.21 oz. and a standard deviation of 0.22, bag~N(1.21, 0.22).

• Suppose the manager takes a sample of 16 bags and observes a sample mean of 1.20 with a standard deviation of 0.20. One of the bags in the sample weighs 1.3 ounces. Calculate a z-score for the value of 1.3 and interpret its meaning.
• The manager asks the following question: “If the mean and standard deviation of the population are true as given (i.e., ? = 1.21 and ? = 0.22) and I took a random sample of 16 bags, what is the probability that the sample mean would be greater than 1.3 oz.?” Note: This problem is a sampling distribution problem. Use the standard error in calculating a z-score.
• The manager asks the same question as in the previous problem, but in reference to 49 bags. He chooses a larger sample size. Compare the probabilities from using a sample of 16 bags and to 49 bags. Explain why the answers are different.