Question: The advertised weight of a package of M&M Peanut Butter candies is 1.63 ounces. However, suppose the actual weights of these candy bars vary according to a normal distribution with mean 1.66 ounces and standard deviation 0.02 ounces. (Make sure to give z-scores and probability values, where appropriate, to justify your answers.)

1. What is the probability that the weight of an individual package is less than or equal to the advertised weight of 1.63 ounces?
2. Suppose you plan to take a simple random sample of five 1.63 -ounce packages of M & M Peanut Butter candies and calculate the sample mean weight. Can you assume that the sampling distribution of the sample means will be approximately normal? Explain.
3. Give the mean and standard deviation of the sampling distribution for samples of size 5 ?
4. Calculate the probability that the sample mean of five packages falls below the advertised weight of 1.63 ounces.
5. How do you expect this probability to change if the sample size were 40 instead of 5? Calculate the probability that a sample mean of a random sample of 40 packages falls below the advertised weight.
6. What if the original population had been skewed instead of normal. Which of the three probabilities that you calculated (from question (a) or (d) or (e)) remains approximately correct? Explain.

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