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The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams.

What is the probability that in a simple random sample of 10 pennies minted after 1982, we obtain a sample mean of at least 2.465 grams?

I found out that the answer is 0.2148, but I don't know how to interpret my result. I know that if I take 100 of these pennies about 21 of them will result in a sample mean of at least 2.465, but when I say this it makes little sense. Am I saying that I am averaging 21 of these pennies to be at least 2.465 grams? Are they averaged randomly?

What's a better interpretation? The probability that a randomly selected penny will have a sample mean of 2.465 grams is 21.48%? When I say this, I still can't make sense of the wording.

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It seems you're missing the crucial information in the very text you typed. (And it sounds like you have been taught a way of thinking about probability that is leading you into confusion - one where you interpret probabilities as small-sample proportions. While it might sort of work on the simplest problems, it can easily mislead you. Drop that way of thinking, it is serving you badly here because - among other things - it's leading you to introduce additional numbers that distract you from the actual numbers.)

What is the probability that in a simple random sample of 10 pennies minted after 1982, we obtain a sample mean of at least 2.465 grams?

  • So you take 10 pennies. Not 100. Not 21. 10 pennies. It's right there in your question!

  • You average the weight of those 10 pennies and record the number of grams for that average.

  • Now do it again. And again, and again and again*, an extremely large (notionally infinite) number of times.

  • you now have a 'population' of average-weights from samples of ten coins.

  • If you count what fraction of those average-weights exceed 2.465$g$, you find that (approximately**) a proportion of 0.2148 of those average weights exceed the mark.

* this is the long-run understanding of probability common in the frequentist interpretation of probability, which is the understanding you would bring to this sort of problem

** (averages of samples of size 10 from an approximately normal parent distribution will themselves still only be approximately normally distributed.)

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