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16 Cards in this Set
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Statistics inference

A process that we can get information and draw conclusion about the population from the sample


Estimation

Estimate the parameters of the population using the statistics.
x_bar > mu; s > sigma 

Point estimator (PE)

Draw inference for an unknown parameter about a single value or a point


Drawbacks about PE

(1) For continuous probability, P(x=a)=0 implies we hava zero prob. to have x_bar = mu, for example.
(2) Most of the time, we want how close the statistics to the parameters. PE cannot provides this info. (3) As sample size (ss) increase PE won't be affect. As a good estimator, if ss increases, it will get more accurate. 

Interval estimator

Draw inference for the unknown parameter about an interval.
i.e. within that interval, the parameter will be included with x% of confidence. 

What is are good qualities for the estimator?

(1) unbiased
(2) consistency (3) relatively efficiency 

unbiased

expected value of an estimator is equal to the parameter


consistency

as sample size increase, the difference btw an estimator and the parameter will be decrease


relative efficiency

If there are 2 unbiased estimator, the one with smaller variance is relative efficiency.


CI of mu, with CI = 1alpha

CI=[x_barza_2x(sigma/r_n),
x_bar+za_2x(sigma/r_n)] interpret: the probability that mu as a fixed value included in the CI is 1alpha 

lower/upper CI limit

LCI = lower limit of CI
UCI = upper limit of CI 

Useful z_a_over_2 values:
90%=? 95%=? ... 
1a z_a_over_2
90% 1.645 95% 1.96 98% 2.33 99% 2.575 

CI interpretation

About CI% of time for the repeatly sampling till infinity time that the true parameter is included within CI
Since in reality, we pick only one sample usually, our statistics and we only know that if we repeatly draw the samples we will have CI% of the samples contain the parameter 

CI is affected by ___, ___, ___

(1) 1alpha i.e. confidence level; inc>narrow
(2) n; inc>narrow (3) sigma; inc> wider the wider CI is, the less precisous info it produces 

sigma of sample median

= 1.2533sigma/root_n
=> bigger than sample mean, less relative efficiency 

Sample size

If width of CI and CI% is specific, and sigma is given then
w = z_a_2*[sigma/root_n] => n=((z_a_2*sigma)/w)^2 Give sample size, we then can get a sample and find sample mean; Then, find CI. 