A city has occupied households. The number of occupied households did not change during the time of two surveys.
Survey A randomly selected households throughout the city and estimated that of all occupied households use public transit at least once per week. The margin of error for Survey A was percentage points.
Survey B independently and randomly selected households throughout the city and estimated that of all occupied households use public transit at least once per week. The margin of error for Survey B was percentage points.
Assuming that both sampling methods were sound and that each stated margin of error describes a plausible range for the population percentage, which choice gives the complete range of numbers of occupied households that is consistent with both survey results?
When two surveys estimate the same population value, first turn each estimate and stated margin of error into a plausible range. If the question asks for values consistent with both surveys, use the intersection: take the greater lower endpoint and the lesser upper endpoint. Only after finding that shared percentage range should you convert percentages to population counts.
Hints
Create a range for each survey
For each survey, subtract its margin of error from the estimate and add its margin of error to the estimate.
Look for values supported by both surveys
The percentage must fall in the overlap of the two plausible percentage ranges, not merely in either range.
Use the total number of households
Convert each endpoint of the overlapping percentage range to a number of households by multiplying by .
Desmos Guide
Identify the overlapping percentage range
An algebraic comparison is faster than graphing here. Determine each survey's lower and upper percentage endpoints, then keep only the overlap between the two ranges.
Calculate the household-count endpoints
In the Desmos expression line, enter 12000*0.37 and 12000*0.42. These calculations give the household counts at the endpoints of the overlapping percentage range.
Step-by-step Explanation
Find each plausible percentage range
For Survey A, subtract and add the -percentage-point margin of error:
For Survey B, subtract and add the -percentage-point margin of error:
Use the overlap of the two ranges
A population percentage consistent with both surveys must be in both ranges. The overlapping range is:
Convert the percentage endpoints to household counts
Find the corresponding numbers of households:
Thus, the complete range is from 4,440 to 5,040 households.