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

A company that produces socks wants to estimate the percent of the socks produced in a typical week that are defective. A random sample of $310$ socks produced in a certain week were inspected. Based on the sample, it is estimated that $12 percent sign$ of all socks produced by the company in this week are defective, with an associated margin of error of $3.62 percent sign$ . Based on the estimate and associated margin of error, which of the following is the most appropriate conclusion about all socks produced by the company during this week?

$3.62 percent sign$ of the socks are defective.

It is plausible that between $8.38 percent sign$ and $15.62 percent sign$ of the socks are defective.

$12 percent sign$ of the socks are defective.

It is plausible that more than $15.62 percent sign$ of the socks are defective.

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Explanation

Choice B is correct. It’s given that, based on the sample, an estimate of $12 percent sign$ of all socks produced by the company in a certain week are defective, with an associated margin of error of $3.62 percent sign$ . This estimate, plus or minus the margin of error, gives an interval of plausible values for the actual percent of all socks produced by the company that week that are defective. Subtracting $3.62 percent sign$ from $12 percent sign$ yields $8.38 percent sign$ . Adding $3.62 percent sign$ to $12 percent sign$ yields $15.62 percent sign$ . Therefore, it is plausible that between $8.38 percent sign$ and $15.62 percent sign$ of all socks produced by the company are defective.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect. $12 percent sign$ is the estimated percent of defective socks based on the sample. However, since the margin of error for this estimate is known, the most appropriate conclusion is not that the percent of defective socks is exactly $12 percent sign$ but instead that it lies in an interval of plausible percents.

Choice D is incorrect and may result from conceptual errors.