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a) Develop a 90% confidence interval for the population mean. b) Develop a 95% confidence interval for the population mean. c) Develop a 99% confidence interval for the population mean. d) Comment on what happened to the margin of error and the confidence interval as the confidence level increased. Ideal Situation: BUT…Typical question … “How big does the sample size, nneed to be?”Answer requires specification of a confidence level and a margin of error.
OIS 2340 –Business Statistics Estimating Population Parameters 7 Determining a Sample Size for Estimating μ when σ is Known:2222zzneee= desired margin of error. Example An advertising company wishes to estimate the mean household income for all single working professionals who own a foreign automobile. If the advertising company wants a 90% confidence interval estimate with a margin of error of $2,500, what sample size is needed if the population standard deviation is known to be $27,500? Determining a Sample Size for Estimating μ When σ is Unknown 1.Use a standard deviation computed from data of previous studies. 2.Use a pilot study to select a preliminary sample and use its standard deviation. 3.Use a judgment or “best guess” for the value of the standard deviation. Often, the range of the data divided by 4 or 6 is used. Conservative (larger) Estimate: max. value min. value4Typical (smaller) Estimate: max. value min. value6
OIS 2340 –Business Statistics Estimating Population Parameters 8 Example The range for a set of data is estimated to be 36. a. What is a conservative planning value for the population standard deviation? b. At 95% confidence, how large a sample would provide a margin of error of 3? c. At 95% confidence, how large a sample would provide a margin of error of 2? Confidence Interval for Population Proportion Confidence Interval = 1pppZnA simple random sample of 400 individuals provides 100 “Yes” responses. Find the 95% confidence interval for the proportion of the population answering “Yes.”point estimate for the population proportion (sample proportion) Z-value that corresponds to the confidence level standard error of the proportion NOTE: Using this estimate for the standard error is valid assuming that np > 5 and n(1 –p) > 5. Example
OIS 2340 –Business Statistics Estimating Population Parameters 9 Sample Size for Estimating Population Proportion2