Sample Size and CI’s for the Population Mean ( 

Sample Size and CI’s for  Suppose we wish to estimate a population mean  using a 95% CI and have a margin of error no larger than E units. What sample size do we need to use? Suppose we wish to estimate a population mean  using a 95% CI and have a margin of error no larger than E units. What sample size do we need to use? Recall the “large” sample CI for  is given by: Recall the “large” sample CI for  is given by: MARGIN OF ERROR (E) Note: The z-value should actually be a t-distribution value, but for sample size planning purposes we will use a standard normal value. 95% z = % z = % z = 2.576

Sample Size and CI’s for  For a 95% CI if we want margin of error, E we have For a 95% CI if we want margin of error, E we have After some wonderful algebraic manipulation After some wonderful algebraic manipulation Oh, oh! We don’t know s !! 1.“Guesstimate” 2.Use sample SD from pilot or prior study. 3.Use fact 95% of observations generally lie with 2 SD’s of the mean thus where Range represents the expected maximum – minimum we would see in sample. Could also use fact that 99% lie within 3 SD’s and use 6 instead of 4 in our crude approximation.

Example: Estimating Mean Cholesterol Level of Females 30 – 40 yrs. of age Q: What sample size would be necessary to estimate the mean cholesterol level for the population of females between the ages of 30 – 40 with a 95% confidence interval that has a margin of error no larger than E = 3 mg/dl?

Sample Size and CI’s for   Suppose from a pilot study we find s = 19.8 mg/dl We can use this estimate to find the sample size that will give E = 3 mg/dl. We can use this estimate to find the sample size that will give E = 3 mg/dl. Standard normal values 90% = % = % = 2.576

Sample Size and CI’s for  Suppose we do not have any information about the standard deviation of the cholesterol levels of individuals in this population. Suppose we do not have any information about the standard deviation of the cholesterol levels of individuals in this population. We could use the Range/4 or Range/6 as crude approximations to the standard deviation. We could use the Range/4 or Range/6 as crude approximations to the standard deviation. What is the smallest serum cholesterol level we would expect to see? 100 mg/dl (my guess) What is the smallest serum cholesterol level we would expect to see? 100 mg/dl (my guess) What is the largest? 300 mg/dl (my guess again) What is the largest? 300 mg/dl (my guess again) SD approximation = 200/4 = 50 mg/dl SD approximation = 200/4 = 50 mg/dl or or SD approximation = 200/6 = mg/dl SD approximation = 200/6 = mg/dl

Sample Size and CI’s for  Using this crude estimate for the standard deviation we find the following sample size requirements Using this crude estimate for the standard deviation we find the following sample size requirements or or