Question 9·Hard·Inference from Sample Statistics and Margin of Error
A researcher interviewed 411 randomly selected US residents and asked about their views on the use of nuclear energy. The results are shown below.
| Response | Frequency |
|---|---|
| Strongly favor | 56 |
| Somewhat favor | 214 |
| Somewhat oppose | 104 |
| Strongly oppose | 37 |
When the survey was given, the population of the United States was 300 million. Based on the sample data, what is the best estimate, in millions, of the difference between the number of US residents who somewhat favor or strongly favor the use of nuclear energy and the number of those who somewhat oppose or strongly oppose it? (Round your answer to the nearest whole number.)
(Express the answer as an integer)
For questions that ask you to estimate a population value from sample data, first combine categories as the question specifies (here, all “favor” responses together and all “oppose” responses together). Find the difference in counts in the sample, then convert that difference to a proportion of the total sample size. Finally, scale that proportion by the given population size and pay close attention to the requested units (such as millions) and rounding instructions. This proportion-based approach is quick, avoids dealing with huge numbers directly, and reduces arithmetic errors.
Hints
Combine like response types
Which responses should be grouped together as “favoring” nuclear energy, and which as “opposing” it? Add the frequencies within each group first.
Compare favoring vs. opposing in the sample
Once you have totals for favoring and opposing, subtract to find how many more people in the sample favor nuclear energy than oppose it.
Use proportions to estimate the population
Take the sample difference you found and divide by 411 to get a proportion. Then think about how to use that proportion to estimate how many people that would be out of 300 million, and remember the answer should be in millions and rounded to the nearest whole number.
Desmos Guide
Compute the scaled difference
In Desmos, type the expression (270-141)/411*300. The output gives the estimated difference in millions of people between those who favor and those who oppose nuclear energy; then round that value to the nearest whole number.
Step-by-step Explanation
Group the responses into favor and oppose
First, combine the categories that count as favoring nuclear energy and those that count as opposing it.
- Favoring: Strongly favor (56) + Somewhat favor (214)
- Total favoring
- Opposing: Somewhat oppose (104) + Strongly oppose (37)
- Total opposing
Find the difference in the sample
Now find how many more people in the sample favor nuclear energy than oppose it.
- Difference in sample
So, in the sample of 411 people, there are 129 more people who favor than oppose nuclear energy.
Turn the sample difference into a proportion
To use the sample to estimate the whole US population, treat the sample as representative and convert the difference to a proportion of the sample.
The sample size is 411, so the proportion difference is
This is approximately (about 31% of the sample).
Scale the proportion to the US population and round
Apply this proportion difference to the entire US population of 300 million:
This value is in millions of people. Rounding to the nearest whole number gives 94, so the best estimate of the difference is 94 million.