Question 42·Hard·Inference from Sample Statistics and Margin of Error
County L has 62,000 households. A random sample of 400 households found that 68 percent reported that their primary internet connection is via fiber-optic cable. The margin of error for this estimate is ±4 percentage points at a 95 percent confidence level.
Which of the following could be the number of households in County L whose primary internet connection is via fiber-optic cable?
On SAT questions involving a sample proportion and a margin of error, first turn the margin of error into a range by adding and subtracting it from the reported percentage. Then, depending on what the question asks, either (1) convert that percentage range into actual counts by multiplying by the total population, or (2) convert each answer choice into a percentage and see whether it falls inside the range. This approach is quick, avoids heavy computation, and keeps your reasoning aligned with the idea of a confidence interval.
Hints
Interpret the margin of error
Focus on the phrase "margin of error ±4 percentage points." What two percentages does that create around the 68% estimate?
Turn percentages into actual numbers of households
Once you know the lowest and highest possible percentages, multiply those percentages (as decimals) by the total number of households, 62,000, to get a range of possible household counts.
Check the answer choices against your range
After you find the minimum and maximum possible numbers of households, see which answer choices fall inside that interval and which fall outside it.
Desmos Guide
Compute the lower and upper bounds in Desmos
In two separate lines, type 0.64*62000 and 0.72*62000. Desmos will display the lower and upper bounds for the possible number of fiber-optic households.
Compare answer choices to the Desmos results
Look at the two numbers Desmos gives you and note the interval they form. Then check which of the four answer choices lies between these two values (inclusive) and which ones fall outside the range.
Step-by-step Explanation
Use the margin of error to find the possible percentage range
The survey result is 68% with a margin of error of ±4 percentage points.
So the possible true percentage of households with fiber-optic is between:
- Lower bound: percent
- Upper bound: percent
That means the true proportion could be anywhere from 64% to 72%.
Convert the percentage range to a range of household counts
County L has 62,000 households.
Find the minimum possible number of fiber-optic households using 64%:
Find the maximum possible number using 72%:
So the plausible number of fiber-optic households is between 39,680 and 44,640.
Compare each answer choice to this range
Now compare each option to the range 39,680 to 44,640:
- 38,000 is less than 39,680
- 40,500 is between 39,680 and 44,640
- 47,000 is greater than 44,640
- 50,000 is greater than 44,640
Only 40,500 lies within the possible range, so the correct answer is 40,500.