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Required Sample Size to Estimate  Within a Specified M argin of Error With a Desired Level of Confidence

Required Sample Size to Estimate  Within a Specified M argin of Error With a Desired Level of Confidence. Chapter 7. Required Sample Size To Estimate a Population Mean .

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Required Sample Size to Estimate  Within a Specified M argin of Error With a Desired Level of Confidence

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  1. Required Sample Size to Estimate  Within a Specified Margin of Error With a Desired Level of Confidence Chapter 7

  2. Required Sample Size To Estimate a Population Mean  • If you desire a C% confidence interval for a population mean  with an accuracy specified by you, how large does the sample size need to be? • We will denote the accuracy by ME, which stands for Margin of Error.

  3. Example: Sample Size to Estimate a PopulationMean  • Suppose we want to estimate the unknown mean height  of male students at NC State with a confidence interval. • We want to be 95% confident that our estimate is within .5 inch of  • How large does our sample size need to be?

  4. Confidence Interval for 

  5. Good news: we have an equation • Bad news: • Need to know s • We don’t know n so we don’t know the degrees of freedom to find t*n-1

  6. A Way Around this Problem: Approximate by Using the Standard Normal

  7. Confidence level Sampling distribution of x .95

  8. Estimating s • Previously collected data or prior knowledge of the population • If the population is normal or near-normal, then s can be conservatively estimated by s  range 6 • 99.7% of obs. within 3  of the mean

  9. Example:samplesize to estimate mean height µ of NCSU undergrad. male students We want to be 95% confident that we are within .5 inch of , so • ME = .5; z*=1.96 • Suppose previous data indicates that s is about 2 inches. • n= [(1.96)(2)/(.5)]2 = 61.47 • We should sample 62 male students

  10. Example: Sample Size to Estimate a PopulationMean -Textbooks • Suppose the financial aid office wants to estimate the mean NCSU semester textbook cost  within ME=$25 with 98% confidence. How many students should be sampled? Previous data shows  is about $85.

  11. Example: Sample Size to Estimate a Population Mean -NFL footballs • The manufacturer of NFL footballs uses a machine to inflate new footballs • The mean inflation pressure is 13.5 psi, but uncontrollable factors cause the pressures of individual footballs to vary from 13.3 psi to 13.7 psi • After throwing 6 interceptions in a recent game, Peyton Manning complains that the balls are not properly inflated. The manufacturer wishes to estimate the mean inflation pressure to within .025 psi with a 99% confidence interval. How many footballs should be sampled?

  12. Example: Sample Size to Estimate a Population Mean  • The manufacturer wishes to estimate the mean inflation pressure to within .025 pound with a 99% confidence interval. How may footballs should be sampled? • 99% confidence  z* = 2.576; ME = .025 •  = ? Inflation pressures range from 13.3 to 13.7 psi • So range =13.7 – 13.3 = .4;   range/6 = .4/6 = .067 . . . 1 2 3 48

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