Question 25·Easy·Inference from Sample Statistics and Margin of Error
During a baseball game with 30,000 occupied seats, an usher wants to estimate how many spectators are wearing hats. The usher randomly selects six seating sections of 200 seats each and counts the number of spectators wearing hats in each selected section.
32, 25, 28, 30, 27, 24
Based on the mean of this sample, which of the following ranges best estimates the total number of spectators wearing hats in the stadium?
For questions where a sample is used to estimate a total for a larger population, first find the sample proportion or average: add the sample counts, divide by the sample size (or total sampled units), and get a percentage or rate. Then multiply that proportion by the total population size to estimate the overall count, and finally match this estimate to the closest answer range. Always pay attention to how the sample is structured (for example, sections of 200 seats) so you use the correct total number of sampled units.
Hints
Combine the sample data
First, find how many spectators in total (across all 6 sections) were wearing hats. What do you get when you add 32, 25, 28, 30, 27, and 24?
Turn the sample into a proportion
How many seats were observed in the sample altogether? Use the total number of hats and the total number of sampled seats to find the fraction of spectators in the sample who wear hats.
Use the proportion to estimate the total
You now have the approximate proportion of spectators wearing hats. How can you use this proportion to estimate how many out of all 30,000 spectators are wearing hats?
Desmos Guide
Compute the estimate directly
In Desmos, type the expression 30000*(32+25+28+30+27+24)/(6*200) and evaluate it. The output is the estimated total number of spectators wearing hats; compare that number to the given answer ranges to see which range it falls in.
Step-by-step Explanation
Add the sample counts and find the sample size
The usher counted hats in 6 sections.
- Add the hat counts:
- spectators wearing hats in the sample.
- Each section has 200 seats, and there are 6 sections, so the total number of sampled seats is:
- seats in the sample.
Find the sample proportion wearing hats
Use the sample to estimate the proportion of spectators wearing hats.
- Proportion in the sample:
- As a decimal, , or about of the sampled spectators wearing hats.
Scale the proportion up to all 30,000 spectators and choose the range
Use the sample proportion to estimate the total number of hat-wearers in the whole stadium.
So we estimate there are about 4,150 spectators wearing hats. This number falls between 4,000 and 4,300, so the best answer is 4,000 to 4,300.