Question

Katrina wants to estimate the proportion of adults who read at least 10 books last year. to do​ so, she obtains a simple random sample of 100 adults and constructs a​ 95% confidence interval. matthew also wants to estimate the proportion of adults who read at least 10 books last year. he obtains a simple random sample of 400 adults and constructs a​ 99% confidence interval. assuming both katrina and matthew obtained the same point​ estimate, whose estimate will have the smaller margin of​ error? justify your answer.

Answer 1

E=Z*sqrt (p(1-p)/N), where E= error margin, p=proportion, N=sample size

Katrina's margin error at 85% confidence interval: E=1.96*sqrt (p(1-p)/100) = 0.196 sqrt (1(1-p))

Mathew's margin error at 99% confidence interval: E= 2.58*sqrt (p(1-p)/400) = 0.129 sqrt (p(1-p))

Since both obtained same estimate of proportion (that is, value of p), it can be seen that Mathew's estimate will have a small error (That is, 0.129 is smaller than 0.196). This can be attributed to larger sample size although a wider confidence (99%) interval was considered.