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Problem-Solving and Data Analysis / Inference from sample statistics and margin of error Difficulty: Medium

A bag containing 10,000 beads of assorted colors is purchased from a craft store. To estimate the percent of red beads in the bag, a sample of beads is selected at random. The percent of red beads in the bag was estimated to be 15%, with an associated margin of error of 2%. If r is the actual number of red beads in the bag, which of the following is most plausible?

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Explanation

Choice B is correct. It was estimated that 15% of the beads in the bag are red. Since the bag contains 10,000 beads, it follows that there are an estimated $10,000 times 0 point 1 5, equals 1,500$ red beads. It’s given that the margin of error is 2%, or $10,000 times 0 point 0 2, equals 200$ beads. If the estimate is too high, there could plausibly be $1,500 minus 200, equals 1,300$ red beads. If the estimate is too low, there could plausibly be $1,500 plus 200, equals 1,700$ red beads. Therefore, the most plausible statement of the actual number of red beads in the bag is $1,300 is less than r, which is less than 1,700$ .

Choices A and D are incorrect and may result from misinterpreting the margin of error. It’s unlikely that more than 1,700 beads or fewer than 1,300 beads in the bag are red. Choice C is incorrect because 200 is the margin of error for the number of red beads, not the lower bound of the range of red beads.