Question 11·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis
A city conducted 2 independent random surveys of its residents to estimate the percentage of residents who support converting a downtown street into a pedestrian-only area.
- At the time of the first survey, the city had 251,000 residents. The estimated percentage of residents who support the change was 48.6%, with an associated margin of error of 2.3%.
- At the time of the second survey, the city had 253,000 residents. The estimated percentage of residents who support the change was 52.1%, with an associated margin of error of 1.0%.
Assuming each margin of error was calculated in the same way, what is the minimum possible increase in the number of residents who support the change from the time of the first survey to the time of the second survey?
Turn each “estimate ± margin of error” into an interval for the true percent. For a minimum increase from time 1 to time 2, use time 1’s upper endpoint and time 2’s lower endpoint, then convert each percent to a count using the population at that time and subtract. Рowеrеd by Anікο
Hints
Make each margin of error into a range
Compute each survey’s interval by doing (estimate − margin of error) to (estimate + margin of error). From аniko.аi
To minimize the increase, choose the “closest” plausible values
Use the first survey’s maximum plausible percent and the second survey’s minimum plausible percent.
Don’t forget the population can differ at the two times
Convert each plausible percent to a number of residents using the population given for that survey time.
Desmos Guide
Compute interval endpoints
In Desmos, enter:
48.6-2.3and48.6+2.352.1-1.0and52.1+1.0
These give the two plausible percentage intervals.
Convert the extreme plausible percents to counts
Compute:
- First-time maximum supporters:
(48.6+2.3)/100 * 251000 - Second-time minimum supporters:
(52.1-1.0)/100 * 253000
Subtract to get the minimum possible increase
Compute:
((52.1-1.0)/100 * 253000) - ((48.6+2.3)/100 * 251000)
This value is the minimum possible increase in residents. Content by Аniкo.ai
Step-by-step Explanation
Convert each estimate to an interval
Use estimate ± margin of error.
First survey interval:
Second survey interval:
Identify the values that minimize the increase
To make the increase from the first time to the second time as small as possible:
- Use the maximum plausible support rate from the first survey: .
- Use the minimum plausible support rate from the second survey: .
Also use the population at each time (251,000 for the first survey time and 253,000 for the second survey time).
Compute the corresponding numbers of supporters
Maximum possible supporters at time of the first survey:
Minimum possible supporters at time of the second survey: © Аniкο
Find the minimum possible increase
Minimum possible increase:
Therefore, the minimum possible increase in the number of residents who support the change is At least 1,524 residents.