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

To estimate the proportion of a population that has a certain characteristic, a random sample was selected from the population. Based on the sample, it is estimated that the proportion of the population that has the characteristic is $0.49$ , with an associated margin of error of $0.04$ . Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of the population that has the characteristic?

It is plausible that the proportion is between $0.45$ and $0.53$ .

It is plausible that the proportion is less than $0.45$ .

The proportion is exactly $0.49$ .

It is plausible that the proportion is greater than $0.53$ .

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Explanation

Choice A is correct. It’s given that the estimate for the proportion of the population that has the characteristic is $0.49$ with an associated margin of error of $0.04$ . Subtracting the margin of error from the estimate and adding the margin of error to the estimate gives an interval of plausible values for the true proportion of the population that has the characteristic. Therefore, it’s plausible that the proportion of the population that has this characteristic is between $0.45$ and $0.53$ .

Choice B is incorrect. A value less than $0.45$ is outside the interval of plausible values for the proportion of the population that has the characteristic.

Choice C is incorrect. The value $0.49$ is an estimate for the proportion based on this sample. However, since the margin of error for this estimate is known, the most appropriate conclusion is not that the proportion is exactly one value but instead lies in an interval of plausible values.

Choice D is incorrect. A value greater than $0.53$ is outside the interval of plausible values for the proportion of the population that has the characteristic.