Question 31·Easy·Inference from Sample Statistics and Margin of Error
A library contains 1,200 books. A librarian randomly selects 40 of these books and finds that 12 of them are science-fiction titles.
Based on this sample, what is the best estimate of the total number of science-fiction books in the library?
For questions that ask you to estimate a total from a random sample, first find the sample proportion (part over whole in the sample), convert it to a decimal or percentage, then multiply that proportion by the total population size. Check that your answer makes sense: it should be in the same rough percentage range as the sample and should not be wildly larger or smaller than the total population.
Hints
Use the sample to find a fraction
From the 40 books chosen, how many were science-fiction? Write this as a fraction of the sample.
Turn the fraction into a percentage or decimal
Convert the fraction you found (science-fiction over total sampled) into a decimal or percentage. This tells you approximately what fraction of the entire library might be science-fiction.
Scale up to the whole library
Take the proportion of science-fiction books from the sample and apply it to all 1,200 books. What operation should you use?
Desmos Guide
Use Desmos to compute the estimate
In Desmos, type the expression (12/40)*1200 and press Enter. The output value is the estimated number of science-fiction books based on the sample proportion.
Step-by-step Explanation
Find the sample proportion of science-fiction books
In the sample of 40 books, 12 are science-fiction.
The sample proportion of science-fiction books is
Simplify this fraction: .
So about of the sampled books are science-fiction.
Apply the proportion to the entire library
Assuming the sample is representative of the whole library, we use the same proportion for all 1,200 books.
Multiply the total number of books by the sample proportion:
Now calculate this product.
Compute the product to get the estimate
Compute :
So the best estimate of the total number of science-fiction books in the library is 360.