Question 54·Hard·Inference from Sample Statistics and Margin of Error
A polling firm surveyed a simple random sample of 400 adult residents of a metropolitan area and found that 52 percent support a proposed public–transit expansion. The firm reports a margin of error of ±5 percentage points for this estimate at the 95 percent confidence level.
If the sample is representative of the entire population of 3.2 million adult residents in the metropolitan area, which interval best represents the possible number of residents who support the expansion?
For margin-of-error questions, first interpret the statistic and margin clearly: a poll result of 52% with a ±5 percentage point margin means the true population percentage is likely between 47% and 57%. Either (1) convert that percentage range to decimals and multiply each by the total population to get a range of counts, or (2) to avoid big multiplications, divide each answer choice’s endpoints by the population to see what percentage range it represents, then pick the one that matches 47% to 57%. Always keep straight that a margin of error in "percentage points" is added to and subtracted from the reported percentage, not from the raw number of people or just the sample size.
Hints
Interpret the margin of error
"Margin of error of ±5 percentage points" around 52% describes a range of possible support percentages. What do you get if you subtract 5 and add 5 to 52?
From percentage to proportion
Once you have the low and high percentages, convert each to a decimal proportion (for example, 25% is 0.25). What are the decimal forms of your two percentages?
From proportion to number of people
You know the total adult population is 3.2 million. How do you use each of the two decimal proportions with 3,200,000 to get a low and high estimate for the number of supporters?
Connect to the choices
After you find the two numbers, compare them with the endpoints in the answer choices and select the matching interval.
Desmos Guide
Enter the expressions for the bounds
In Desmos’s calculator, type these two expressions on separate lines:
(0.52 - 0.05) * 3200000(0.52 + 0.05) * 3200000
These represent the lower and upper estimates for the number of supporters.
Read the numerical outputs
Look at the two numeric results Desmos gives. The smaller value is the lower bound and the larger value is the upper bound of the interval. Compare these two numbers to the answer choices and select the interval that matches them.
Step-by-step Explanation
Use the margin of error to find the percentage range
The poll estimates that 52% of adults support the expansion, with a margin of error of ±5 percentage points at the 95% confidence level.
That means the true percentage of supporters in the whole population could be as low as
and as high as
- .
So the possible percentage of residents who support the expansion is from 47% to 57%.
Convert percentages to decimal form
To turn percentages into decimals, divide by 100.
- becomes
- becomes
So the possible proportion (fraction) of the population that supports the expansion is from to .
Set up expressions for the number of residents
There are 3.2 million adult residents in the metropolitan area. As a number, that is
To find the possible number of supporters, multiply each endpoint of the proportion interval by the total population:
- Lower bound:
- Upper bound:
These two products give the smallest and largest plausible numbers of residents who support the expansion.
Compute the bounds and match to an answer choice
Now compute the products:
- Lower bound:
- Upper bound:
So the interval for the possible number of adult residents who support the expansion is 1,504,000 to 1,824,000, which corresponds to choice B.