Question 24·Hard·Inference from Sample Statistics and Margin of Error
A polling organization surveyed a simple random sample of 400 registered voters in County Z. The survey found that 248 of those sampled said they support Proposition 12. The organization reported the result as percentage points, where percentage points is the margin of error for a confidence level.
County Z has 52,000 registered voters. Based on the organization’s report, which of the following values is not within the confidence interval for the number of registered voters in County Z who support Proposition 12?
For margin-of-error questions, first convert the sample result into a proportion, then apply the margin of error to form a proportion interval (using plus and minus). Convert percentages to decimals carefully. If the question asks about actual counts in a known population, multiply both ends of the proportion interval by the population size to get a count interval, then simply check which answer choice lies outside that range. This step-by-step approach reduces mistakes with percent vs. decimal and keeps your work organized.
Hints
Start with the proportion in the sample
First, turn the 248 out of 400 sampled voters into a proportion or percentage. What fraction or percent of 400 is 248?
Use the margin of error correctly
The report says percentage points. Find the lower and upper percentages by subtracting and adding 4.8 to 62, then convert these percentages into decimals.
Translate percent interval to number-of-voters interval
Once you have the lower and upper bounds for the proportion, multiply each by 52,000 to get the corresponding lower and upper bounds for the number of voters who support Proposition 12.
Compare answer choices to your interval
After you find the approximate minimum and maximum possible numbers of supporters, check which of the choices lies outside that range.
Desmos Guide
Compute the confidence interval endpoints for the number of voters
In Desmos, type these two expressions on separate lines:
(0.62 - 0.048)*52000(0.62 + 0.048)*52000
These give the lower and upper bounds of the 95% confidence interval for the number of supporters.
Compare answer choices to the interval
Next, enter each choice as its own expression (for example, 30100, 32000, 34500, 36000). Compare these values to the two bounds you calculated and identify which one lies outside the interval between the lower and upper bounds.
Step-by-step Explanation
Find the sample proportion that supports Proposition 12
The poll found that 248 out of 400 sampled voters support Proposition 12.
So the sample proportion is
This is the reported 62%.
Apply the margin of error to get the confidence interval for the proportion
The organization reported percentage points.
That means the true proportion is estimated to be between:
- Lower bound:
- Upper bound:
Convert these percentages to decimals:
So the 95% confidence interval for the proportion is from to .
Convert the proportion interval to a voter-count interval
County Z has 52,000 registered voters. To estimate the number of supporters, multiply each end of the proportion interval by 52,000.
Lower bound:
Upper bound:
So the 95% confidence interval for the number of supporters is from about 29,744 to 34,736.
Compare the answer choices to this interval
We now check which choices fall outside 29,744 to 34,736:
- 30,100 is between 29,744 and 34,736 (inside the interval).
- 32,000 is between 29,744 and 34,736 (inside the interval).
- 34,500 is between 29,744 and 34,736 (inside the interval).
- 36,000 is greater than 34,736 (outside the interval).
The value that is not within the 95% confidence interval is 36,000.