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A typical adult has an average IQ score of $105$with a standard deviation of $20$. If $20$randomly selected adults are given an IQ test, what are the probability that the sample mean scores will be between $85$ and $125$ points?

Certain coins have an average weight of $5.201$grams with a standard deviation of $0.065$g. If a vending machine is designed to accept coins whose weights range from $5.111$g to $5.291$g, what is the expected number of rejected coins when $280$randomly selected coins are inserted into the machine?