Download Notes on Confidence Intervals and Choice of Sample Size | STAT 312 and more Study notes Mathematical Statistics in PDF only on Docsity! Stat 312 Fall 2003 09/25/2003 Discussion 4 Confidence Intervals A 100(1-α)% confidence interval for the mean µ of the normal population when the value of σ is know is given by ( x̄− zα/2 · σ√ n , x̄ + zα/2 · σ√ n ) Choice of Sample Size The general formula for the sample size n necessary to ensure that the resulting 100(1-α)% confidence interval has a width of w is obtained from w = 2 · zα/2 · σ/ √ n as n = ( 2 · zα/2 · σ w )2 Example: Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation .75. a. Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85. b. Compute a 98% CI for true average porosity of another seam based on 16 specimens with a sample average porosity of 4.56. c. How large a sample size is necessary if the width of the 95% interval is to be .40? d. What sample size is necessary to estimate true average porosity to within .2 with 99% confidence? A Large Sample Interval for µ If n is sufficient large, the large-sample confidence interval for µ with confidence level approximately 100(1-α)% is x̄± zα/2 · s√ n A Confidence Interval for a Population Proportion A confidence interval for a population proportion p with confidence level approximately 100(1-α)% is p̂ + z2α/2 2n ± zα/2 √ p̂q̂ n + z2 α/2 4n2 1 + (z2α/2)/n One-Sided Confidence Intervals A large-sample upper confidence bound for µ is µ < x̄ + zα · s√ n and a large-sample lower confidence bound for µ is µ > x̄− zα · s√ n Example: A random sample of 539 households from a certain midwestern city was selected, and it was determined that 133 of these households owned at least one firearm (”The social determinants of Gun Ownership: Self- Protection in an Urban Environment,” Criminology, 1997: 629-640). Using a 95% confidence level, calculate a lower confidence bound for the proportion of all households in this city that own at least one firearm. Example: A random sample of 110 lightning flashes in a certain region resulted in a sample average radar echo duration of .81 sec and sample standard deviation of .34 sec (”Lightning Strikes to an Airplane in a Thunderstorm,” J. of Aircraft, 1984: 607-611). Calculate a 99% (two-sided) confidence interval for the true average echo duration µ. Ting-Li Lin
