A consumer group tested random samples of two brands of... | SAT Inference from sample statistics and margin of error Question, Answer Explained

Question bank / Inference from sample statistics and margin of error

Math · Problem-Solving and Data AnalysisMedium

A consumer group tested random samples of two brands of batteries. For brand X, the mean battery life was $41.2$ hours, with a margin of error of $1.5$ hours. For brand Y, the mean battery life was $39.6$ hours, with a margin of error of $1.8$ hours. Which of the following is the most appropriate conclusion?

Brand X batteries have a greater mean life than brand Y batteries, because $41.2$ is greater than $39.6$ .

Every brand X battery lasts longer than every brand Y battery.

Brand Y batteries have a greater mean life than brand X batteries, because brand Y has the larger margin of error.

The samples do not provide convincing evidence that one brand has a greater mean battery life than the other.

Show answer and explanation

Answer: D

Plausible values for brand X's mean are $41.2 \pm 1.5$ , or $39.7$ to $42.7$ hours. Plausible values for brand Y's mean are $39.6 \pm 1.8$ , or $37.8$ to $41.4$ hours. The two intervals overlap, and both include $40$ hours, so it's plausible that the two brands have the same mean battery life.

Choice A compares only the sample means and ignores the margins of error, which show the difference could be due to sampling variability.

Why the other choices are wrong

A. This ignores the margins of error. The intervals 39.7 to 42.7 and 37.8 to 41.4 overlap.
B. The study describes mean battery life, not every individual battery.
C. A larger margin of error means less precision. It says nothing about which mean is greater.

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