Inference from sample statistics and margin of error

From a population of `50000` people, `1000` were chosen at random and surveyed about a proposed piece of legislation. Based on the survey, it is estimated that of people in the population support the legislation, with an associated margin of error of `3%`. Based on these results, which of the following is a plausible value for the total number of people in the population who support the proposed legislation?

A) `350`

B) `650`

C) `16750`

D) `31750`

Answer: C

It is given that an estimated `35%` of people in the population support the legislation, with an associated margin of error of `3%`. Subtracting and adding the margin of error from the estimate gives an interval of plausible values for the true percentage of people in the population who support the legislation. Therefore, it’s plausible that between `32%` and `38%` of people in this population support the legislation. The corresponding numbers of people represented by these percentages in the population can be calculated by multiplying the total population, `50000`, by `0.32` and by `0.38`, which gives `50000xx(0.32) = 16000` and `50000xx(0.38) = 19000`, respectively. It follows that any value in the interval `16000` to `19000` is a plausible value for the total number of people in the population who support the proposed legislation. Of the choices given, only `16750` is in this interval.