Using R for Introductory Statistics : John Verzani

Confidence intervals 1877.3 Confidence intervals for the population mean, µThe success of finding a confidence interval for p in terms of depended on knowing thesampling distribution of once we standardized it. We can use the same approach to finda confidence interval for µ, the population mean, from the sample meanFigure 7.4 Simulation of samplingdistribution of T with n=5. Densitiesof normal distribution and t-distribution are drawn on thehistogram to illustrate that thesampling distribution of T has longertails than the normal distribution.For a random sample X 1 , X 2 , …, X n , the central limit theorem and the formu-las for themean and standard deviation of tell us that for large nwill have an approximately normal distribution. This implies, for example, that roughly95% of the time Z is no larger than 2 in absolute value. In terms of intervals, this can beused to say that µ is in the random interval with probability 0.95.However, σ is usually not known. The standard errror,replaces theunknown a by the known sample standard deviation, s. ConsiderAgain, as the central limit theorem still applies, T has a sampling distribution that isapproximately normal when n is large enough. This fact can be used to constructconfidence intervals such as a 95% confidence interval of