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Sample Size and CI’s for the Population Mean ( m) and the Population Proportion (p)

Sample Size and CI’s for the Population Mean ( m) and the Population Proportion (p). Sample Size and CI’s for m. Suppose we wish to estimate a population mean m using a 95% CI and have a margin of error no larger than E units. What sample size do we need to use?

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Sample Size and CI’s for the Population Mean ( m) and the Population Proportion (p)

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  1. Sample Size and CI’s for the Population Mean (m)and the Population Proportion (p)

  2. Sample Size and CI’s for m • Suppose we wish to estimate a population mean m using a 95% CI and have a margin of error no larger than Eunits. What sample size do we need to use? • Recall the “large” sample CI for mis given by: 95% z = 1.96 90% z = 1.645 99% z = 2.576 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.

  3. Sample Size and CI’s for m • For a 95% CI if we want margin of error, E we have • After some wonderful algebraic manipulation • Oh, oh! We don’t know s !! • “Guesstimate” • Use sample SD from pilot or prior study. • 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.

  4. 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?

  5. Sample Size and CI’s for m • 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. Standard normal values90% = 1.64595% = 1.96099% = 2.576

  6. Sample Size and CI’s for m • 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. • 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) SD approximation = 200/4 = 50 mg/dl or SD approximation = 200/6 = 33.33 mg/dl

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

  8. Sample Size and CI’s for p • Suppose we wish to estimate p using a 95% CI and have a margin of error of 3%. What sample size do we need to use? • Recall the CI for p is given by: MARGIN OF ERROR (E)

  9. Sample Size and CI’s for p • Here for a 95% CI we want E = .03 or 3% • After some wonderful algebraic manipulation • Oh, oh! We don’t know p-hat !! • “Guesstimate” • Use p-hat from pilot or prior study. • Largest n we would ever need comes when p-hat = .50.

  10. Sample Size and CI’s for p • Informed approach • Conservative approach (i.e. worst case scenario) Standard normal values90% = 1.64595% = 1.96099% = 2.578

  11. Sample Size and CI’s for p • Original Question: Suppose we wish to estimate p using a 95% CI and have a margin of error of 3%. What sample size do we need to use? • Assume that we estimate the 5 yr. survival rate for a new kidney cancer therapy, and we know historical that it this survival rate is around 20%. • Using informed approach

  12. Sample Size and CI’s for p • Original Question: Suppose we wish to estimate p using a 95% CI and have a margin of error of 3%. What sample size do we need to use? • Assume that we estimate the 5 yr. survival rate for a new kidney cancer therapy, and we know historical that it this survival rate is around 20%. • Using conservative approach This is why in media polls you they usually report a sampling error of + 3% and that the poll was based on a sample of n = 1000 individuals.

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