Point Estimate|unbiased estimator|Confidence-Interval Estimate
8.1 Estimating a Population Mean
Point Estimate
estimate a single number, or point.
因为:the mean of the sample mean equals the population mean (μx¯ = μ).
所以:In other words, on average, the sample mean equals the population mean.(即mean of sample mean 有机会等于 population mean) For this reason, the sample mean is called an unbiased estimator of the population mean.
当二者相等时:sample mean(一个变量名称) 称为 population mean的unbiased estimator
否则,biased estimator
eg:
则称sample mean 为population mean 的无偏估计。
Confidence-Interval Estimate
在已知 population standard deviation 的情况下,每一个sample 都有一组sample mean和置信区间,对于sample mean的分布来说,无论sample是何分布,sample mean 都是正态分布。
得到一个sample mean(点估计) +已知的population standard deviation,可以模拟一个mean distribution(令mean of mean= sample mean),依据the 68.26-95.44-99.74 rule,可以得到一个区间,若x满足该mean distribution,则x有95.44%的概率落入该区间。因为与mean distribution只有一个mean 不同,所以可以认为所有sample distribution是滑动的mean distribution,当mean of mean 在sample的置信区间时,sample mean 也在mean of mean的置信区间。所以,最后的意思实际上是,我做了n次sample ,得到n个sample mean ,这n个mean中,有0.9544n个值在mean distribution的置信区间中。
所以,在得到n此sample之后,可以取任意置信度a,而位于真实mean of mean的置信度a范围内的sample mean 占总sample mean的a
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