Several pickers are each asked to gather 30 ripe apples and put them in a bag.

(a) Would you expect all of the bags to weigh the same? For one bag, let \(X_{1}\) be the weight of the first apple, \(X_{2}\) the weight of the second apple, and so on. Relate the weight of this bag,

\[\sum_{i=1}^{30} X_{i}\]

to the approximate sampling distribution of \(\bar{X}\).

(b) Explain how your answer to part

(a) leads to the sampling distribution for the variation in bag weights.

(c) If the weight of an individual apple has mean \(\mu=0.2\) pound and standard deviation \(\sigma=\) 0.03 pound, find the probability that the total weight of the bag will exceed 6.2 pounds.