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Difficulty: MediumStatistical Sampling and Studies

A research group conducted a survey using a random sample of 250250 residents of a town with a total population of 18,00018,000 residents. Of the residents surveyed, 64%64\% reported that they recycle regularly. The survey has a margin of error of 3%3\%. Based on the survey results, what is the difference between the maximum and minimum estimated number of residents in the town who recycle regularly?

Answer: 1080

Answer

The correct answer is 1,080.
The sample proportion is 64%64\% with a margin of error of 3%3\%, which establishes an interval of 61%61\% to 67%67\% for the proportion of the population that recycles regularly. Multiplying these boundaries by the total population of 18,00018,000 yields a minimum estimate of 0.61×18,000=10,9800.61 \times 18,000 = 10,980 residents and a maximum estimate of 0.67×18,000=12,0600.67 \times 18,000 = 12,060 residents. The difference between the maximum and minimum estimated number of residents is 12,060−10,980=1,08012,060 - 10,980 = 1,080. Alternatively, this difference can be found by multiplying the total width of the interval (2×0.03=0.062 \times 0.03 = 0.06) by the population size (0.06×18,000=1,0800.06 \times 18,000 = 1,080).

Step-by-Step Solution

1
Determine the range of the proportion of residents who recycle regularly.
The proportion ranges from 61%61\% (0.610.61) to 67%67\% (0.670.67).
The margin of error of 3%3\% is subtracted from and added to the sample proportion of 64%64\%.
2
Calculate the minimum and maximum estimated number of residents who recycle regularly.
The minimum estimate is 10,98010,980 residents and the maximum estimate is 12,06012,060 residents.
Multiply the minimum and maximum proportions by the total population of 18,00018,000: 0.61×18,000=10,9800.61 \times 18,000 = 10,980 and 0.67×18,000=12,0600.67 \times 18,000 = 12,060.
3
Find the difference between the maximum and minimum estimated values.
12,060−10,980=1,08012,060 - 10,980 = 1,080 (or alternatively, 2×0.03×18,000=1,0802 \times 0.03 \times 18,000 = 1,080).
Subtracting the minimum estimate from the maximum estimate yields the difference.

Key Concept

Using sample statistics and margin of error to estimate population parameters.
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Topics

Statistical Sampling and Studies

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