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Problem-Solving and Data Analysis Difficulty: Medium

Based on a random sample from a population, a researcher estimated that the mean value of a certain variable for the population is $20.5$ , with an associated margin of error of $1$ . Which of the following is the most appropriate conclusion?

It is plausible that the actual mean value of the variable for the population is between $19.5$ and $21.5$ .

It is not possible that the mean value of the variable for the population is less than $19.5$ or greater than $21.5$ .

Every value of the variable in the population is between $19.5$ and $21.5$ .

The mean value of the variable for the population is $20.5$ .

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Explanation

Choice A is correct. It's given that based on a random sample from a population, the estimated mean value for a certain variable for the population is $20.5$ , with an associated margin of error of $1$ . This means that it is plausible that the actual mean value of the variable for the population is between $20.5 minus 1$ and $20.5 plus 1$ . Therefore, the most appropriate conclusion is that it is plausible that the actual mean value of the variable for the population is between $19.5$ and $21.5$ .

Choice B is incorrect. The estimated mean value and associated margin of error describe only plausible values, not all the possible values, for the actual mean value of the variable, so this is not an appropriate conclusion.

Choice C is incorrect. The estimated mean value and associated margin of error describe only plausible values for the actual mean value of the variable, not all the possible values of the variable, so this is not an appropriate conclusion.

Choice D is incorrect. Since $20.5$ is the estimated mean value of the variable based on a random sample, the actual mean value of the variable may not be exactly $20.5$ . Therefore, this is not an appropriate conclusion.