We encounter a variety of statistics on a daily basis. Journalists report polling data on upcoming elections, advertisers tell us how many people prefer their products over competitors, and healthcare researchers tell us the prevalence of a given disease across the country. Responsible researchers remind us that these reported statistics fall within a margin of error, but the size of that margin can vary greatly from study to study.
Recall that we can calculate the minimum sample size necessary for a study if we know the desired confidence level, the acceptable margin of error, and the population proportion. (Alternately, we can take a conservative approach and use 0.5 for the proportion.)
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Question 1
What kinds of factors do you think might influence a researcher’s decision about what margin of error is acceptable?
Question 2
The manager of the bookstore at a large university is planning for a new semester and must decide how many textbooks to stock in the store. The manager has noticed that an increasing proportion of students have been buying their books online instead of in the bookstore, so the manager decides to survey a random sample of students to estimate how many students will buy their books online this semester. The manager needs to determine how many students to survey.
- The manager has no idea how many students actually buy their books online. What value should the manager use for [latex]p[/latex] in the sample size calculation?
- Explain why that value will give the manager the best sample size.
- The manager decides to use a confidence level of 95% and a margin of error of 8%. What are the implications of that margin of error? Why might the manager avoid a smaller margin of error? Why might the manager avoid a larger margin of error?
- Go to https://dcmathpathways.shinyapps.io/Inference_prop/ and use the DCMP Inference for a Population Proportion tool to calculate the necessary sample size for the bookstore’s survey.
Question 3
An ecologist is studying the impact of pesticides on honey bees and plans an ecological survey to determine what proportion of bees have pesticides present in their bodies. The ecologist decides to use a confidence level of 95% and a margin of error of 5%, but they do not have an estimated value for the sample proportion. Use the web tool to calculate the necessary sample size for the ecological survey.
Question 4
A biotech company is developing a new rapid test for influenza. After completing their own testing, they claim that their test is correct 97% of the time, with a margin of error of 1% and a confidence level of 95%. An independent researcher will now conduct another study to verify these results. Use the web tool to calculate the necessary sample size for the second study.
Question 5
An auto parts company is conducting routine quality control testing of the airbags produced for new cars. Past testing showed that about 2% of the airbags were defective. The company decides to use a 99% confidence level and wants a margin of error of 1%. Use the web tool to determine how many airbags the company should test as part of the current quality control audit.
Question 6
Now, consider the context you researched before class.
- What margin of error do you think would be acceptable in answering your question? Explain.
- Using a 95% confidence level and a sample proportion of 0.5, calculate the minimum sample size needed to achieve your desired margin of error.
- Compare your answers with a classmate’s. What factors affected the similarities and differences you saw? Would you have used the same margin of error that your classmate chose? Explain.
- Did your Internet source provide a margin of error or sample size for the sample proportion reported? If so, how did those values compare to yours? If not, how does that impact your confidence in the reliability of the information presented?
