In a survey with a given margin of error, the sample proportion estimates the percentage for the whole population. Divide yes responses by people surveyed in Desmos, then subtract the percentage-point margin to find the lower endpoint of the 95% range. A “more than” claim is supported only if that endpoint exceeds the cutoff; the sample percentage alone isn't a guaranteed minimum.
Hints
- Hint 1
The sample proportion is the fraction of surveyed employees who said yes. Its denominator is everyone surveyed, not everyone at the company. Which group size should go under the number who preferred the change?
- Hint 2
A margin of error in percentage points is subtracted directly from the sample percentage. To check whether more than a certain percentage of all employees prefer the change, find the lowest percentage the survey supports.
- Hint 3
A confidence interval is a range of plausible percentages for the whole company. If even its lower endpoint is above a proposed cutoff, the survey supports a claim of more than that cutoff.
Step-by-step
Check the lower end of the survey's range
Step 1Find the sample percentage
The sample proportion is the fraction of surveyed employees who preferred the change. Type in Desmos: said yes out of the surveyed. Desmos shows , or . The denominator is , not ; this percentage describes the surveyed employees.
- Step 2
Find the lowest supported percentage
The margin of error puts the population percentage within percentage points of the sample estimate at the stated confidence level. To check a “more than” claim, use the lower endpoint. Since percentage points is as a decimal, type on the next Desmos line. It shows , or . Don't take of ; percentage points are subtracted directly.
- Step 3
Test the claim against the lower endpoint
The lower endpoint is , which exceeds . To support “more than” a cutoff, even the low end of the range must clear it. So the company can be 95% confident that more than 60% of all employees prefer a four-day workweek. Choice C.