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Stat 31 January 30, 2006 Stat 31 – estimating and testing means Assume a population of numerical values with mean and standard deviation , and assume a random sample of size n consisting of values x1, x2, …, xn with statistics n x x1 xn n and s x x i 1 n 1 2 . In a collection of such samples, the values of the sample mean x are roughly normally distributed with mean and standard deviation (called standard error) SE = . n Therefore, x is an unbiased estimator of . (For the sampling distribution to be normal requires that n be large --- conventionally, n ≥ 30 --- or that the population values themselves be normal.) It follows that x Z n has a standard normal distribution. Alternatively, if is not known, one may substitute the estimator s for to obtain the statistic x T s n which (under strong assumptions) has a t distribution with n – 1 degrees of freedom. When n ≥ 30 this is very close to the standard normal distribution; in this case s is a very good estimator of and the T statistic is very similar to the Z statistic. Confidence intervals for may be constructed as or x t / 2,n1 s x z / 2 n n where z / 2 is such that the fraction /2 of the standard normal distribution falls above z / 2 . (Here, 1 – is the confidence level; for confidence 95% choose = 0.05 and z / 2 1.9600 .) The value t / 2, n 1 plays the same role for the t distribution. To test H0: = 0 vs. HA: ≠ 0, compute x x ( or ) Z T / n s/ n and reject H0 if Z > z / 2 or Z < z / 2 (or for T use t / 2, n 1 ). To test H0: = 0 vs. HA: ≠ 0 (for some conjectured value 0), use the test statistics x 0 x 0 ( or T ). Z s/ n / n (end) 1