Question:

A survey of 625 randomly selected students was conducted to determine student opinion about music. The survey reported that 36 percent of the students surveyed preferred country/western music. The survey estimate had a margin of error of 4 percentage points. A margin of error is reported because

(A) Sample proportions vary from sample to sample.
(B) Students may intentionally respond incorrectly.
(C) Students may misunderstand the survey questions.
(D) The people doing the survey may have recorded results incorrectly.

Level:
Advanced

### Standards

S-IC.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

### Correct answer and commentary

The correct answer to this question is Option (A). Random sampling is at the heart of many contemporary statistical activities. A key statistical outcome for students is that they believe that surveys that are conducted well can provide valid claims about a population even if they only use a random sample that is quite small in comparison to the population (e.g. 1,000 adults in a country relative to the population of that country). The results of such surveys will not provide the exact value for the population and, due to the fact that different samples will result in different responses, this variability must be quantified in some way.

Many students are familiar with the margin of error because it is often reported by the media when discussing polls. (This was not always the case, and that the media report both a measure of center and a measure of variability with poll results is a sign of increased statistical sophistication in our population.) What the margin of error actually represents, however, is not always clear to students. Option (A), indicates that the margin of error accounts for sampling variability and not for some sort of systematic error in the survey. While the remaining options all reflect potential problems with any given survey, such errors are quite insidious because they can be difficult to detect and account for. The margin of error only accounts for variability due to random sampling.