Statistics for Business and Economics 6 th Edition
Statistics for Business and Economics 6 th Edition Chapter 9 Estimation: Additional Topics Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -1
Sample Size Determination Determining Sample Size For the Mean For the Proportion Determine the required sample size to estimate a mean or proportion within a specified margin of error Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -2
Margin of Error § The required sample size can be found to reach a desired margin of error (ME) with a specified level of confidence (1 - ) § The margin of error is also called sampling error § the amount of imprecision in the estimate of the population parameter § the amount added and subtracted to the point estimate to form the confidence interval Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -3
Sample Size Determination Determining Sample Size For the Mean Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Margin of Error (sampling error) Chap 9 -4
Sample Size Determination (continued) Determining Sample Size For the Mean Now solve for n to get Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -5
Sample Size Determination (continued) § To determine the required sample size for the mean, you must know: § The desired level of confidence (1 - ), which determines the z /2 value § The acceptable margin of error (sampling error), ME § The standard deviation, σ Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -6
Example 9. 7: Length of Metal Rods (Sample Size) The lengths of metal rods produced by an industrial process are normally distributed with a standard deviation of 1. 8 millimeters. Based on a random sample of 9 observations from this population, the 99% confidence interval 194. 65 < μ < 197. 75 was found for the population mean length. Suppose that a production manager believes that the interval is too wide for practical use and instead requires a 99% confidence interval extending no further than 0. 50 mm on each side of the sample mean. How large a sample is needed to achieve such an interval? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -7
Answer: ME = 0. 5, σ = 1. 8, zα/2 = z 0. 005= 2. 576 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -8
Required Sample Size Example If = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence? So the required sample size is n = 220 (Always round up) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -9
Sample Size Determination Determining Sample Size For the Proportion Margin of Error (sampling error) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -10
Sample Size Determination (continued) Determining Sample Size For the Proportion cannot be larger than 0. 25, when = 0. 5 Substitute 0. 25 for and solve for n to get Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -11
Sample Size Determination (continued) § The sample and population proportions, and P, are generally not known (since no sample has been taken yet) § P(1 – P) = 0. 25 generates the largest possible margin of error (so guarantees that the resulting sample size will meet the desired level of confidence) § To determine the required sample size for the proportion, you must know: § The desired level of confidence (1 - ), which determines the critical z /2 value § The acceptable sampling error (margin of error), ME § Estimate P(1 – P) = 0. 25 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -12
Required Sample Size Example How large a sample would be necessary to estimate the true proportion defective in a large population within ± 3%, with 95% confidence? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -13
Required Sample Size Example (continued) Solution: For 95% confidence, use z 0. 025 = 1. 96 ME = 0. 03 Estimate P(1 – P) = 0. 25 So use n = 1068 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 9 -14
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