Unit 2: statistical estimation

Case II. Large sample confidence interval for the population mean and the distribution of the population from which the sample is drawn is unknown.

1. 
   It is known, if the sample size is large say n > 30, then the distribution of the static is expected to follow the standard normal variate (central limit theorem), thus the (1-) 100% confidence interval is given by
   

Z_/2 for  is known

2. 
   If  is not known, estimating the population standard deviation by the sample standard deviation S = , then the (1-) 100% confidence interval for the population mean is given by Z_/2
   

Case III. Small sample confidence interval for the population mean: Sampling from a normal distribution and ² is unknown and n < 30.
In this case estimating the population variance ² by the sample variance S² = , the static follows a student ‘t’ distribution with n-1 degree of freedom. i.e.
~ t_n-1 and as the t –distribution is also symmetric as the normal distribution, the (1-) 100% confidence interval for the population in this case is given by:
t_/2^(n-1) . i.e. is expected in the interval ( - t_/2^(n-1) ,
+ t_/2^(n-1) ) with a probability of 1-.


Case IV. Finite population,  is not known and n is large > 30.

In this case, the population variance ² is estimated by the sample variance S² and the standard error of the mean is given by where the term is called the finite population correction and the (1-) 100% confidence interval for the population mean is given by:

Z_/2{ } - means sample variance