ESTIMATINGAPPROPRIATE SAMPLE

SIZE

Determining sample Size

z* σ

√n≤m

How Many monkeys?Researchers would like to estimate the mean cholesterol level ℳ of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1mg/dl of blood of the true value of ℳ at 95% confidence level. A previous study involving this variety of monkeys suggests that the sd of cholesterol level is about σ = 5mg/dl. Obtaining monkeys is time consuming and expensive so the researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate.

CI: 95%z* = 1.96σ= 5mg/dl

z* σ

√n≤ m

1.965

√n≤1

(1.96)≧

1

(5)√n

√n ≧ 9.8

n ≧ 96.04

Researchers would need 97

monkeys to estimate the cholesterol

levels to their satisfaction.

z* σ

√n≤ E

E = maximum error of estimateE = maximum error of estimate

Finding the appropriate Finding the appropriate sample size (n)sample size (n)Mr. Delton asks Charlii to estimate the Mr. Delton asks Charlii to estimate the average age of the students in BHS. Mr. average age of the students in BHS. Mr. Delton is confident that Charlii will be able to Delton is confident that Charlii will be able to find the minimum number of students he find the minimum number of students he needs for his estimate to be reliable. Charlii needs for his estimate to be reliable. Charlii would like to be 99% confident that the would like to be 99% confident that the estimate should be accurate within 1 year. estimate should be accurate within 1 year. From a previous study, the standard deviation From a previous study, the standard deviation of the ages is known to be 3 years. of the ages is known to be 3 years.

Z* = 2.58Z* = 2.58E = 1 yearE = 1 year∂ ∂ = 3 years= 3 years

z* σ

√n≤ E

Z* = 2.58Z* = 2.58E = 1 yearE = 1 year∂ ∂ = 3 years= 3 years

n ≥ 59.9 n ≥ 59.9 Therefore, Charlii needs to have at Therefore, Charlii needs to have at least 60 students to be 99% least 60 students to be 99%

confident that the estimate is confident that the estimate is within 1 year of the true mean age within 1 year of the true mean age

of the students in BHSof the students in BHS

Your Turn!Your Turn!A restaurant owner wishes to find A restaurant owner wishes to find 99% confidence interval of the 99% confidence interval of the true mean cost of a dry martini. true mean cost of a dry martini. How large should the sample be if How large should the sample be if she wishes to be accurate within she wishes to be accurate within $0.10? A previous study showed $0.10? A previous study showed that the population standard that the population standard deviation of the the price was deviation of the the price was $0.12.$0.12.

Facts about the margin of error

z* gets smaller. The trade-off: to obtain smaller margin of error from the same data, you must be willing to accept lower confidence.

σ gets smaller: its easier to pin down ℳ when σ is small

n gets bigger: increasing the sample size reduces the margin of error. To cut the margin of error in half, you must take four times as many observations from the population.