Editing Point estimation (section)

== Point estimate v.s. confidence interval estimate ==
[[File:Point estimation and confidence interval estimation.png|thumb|Point estimation and confidence interval estimation.]]
There are two major types of estimates: point estimate and [[Interval estimation|confidence interval estimate]]. In the point estimate we try to choose a unique point in the parameter space which can reasonably be considered as the true value of the parameter. On the other hand, instead of unique estimate of the parameter, we are interested in constructing a family of sets that contain the true (unknown) parameter value with a specified probability. In many problems of statistical inference we are not interested only in estimating the parameter or testing some hypothesis concerning the parameter, we also want to get a lower or an upper bound or both, for the real-valued parameter. To do this, we need to construct a confidence interval.

[[Confidence interval]] describes how reliable an estimate is. We can calculate the upper and lower confidence limits of the intervals from the observed data. Suppose a dataset x<sub>1</sub>, . . . , x<sub>n</sub> is given, modeled as realization of random variables X<sub>1</sub>, . . . , X<sub>n</sub>. Let θ be the parameter of interest, and γ a number between 0 and 1. If there exist sample statistics L<sub>n</sub> = g(X<sub>1</sub>, . . . , X<sub>n</sub>) and U<sub>n</sub> = h(X<sub>1</sub>, . . . , X<sub>n</sub>) such that P(L<sub>n</sub> < θ < U<sub>n</sub>) = γ for every value of θ, then (l<sub>n</sub>, u<sub>n</sub>), where l<sub>n</sub> = g(x<sub>1</sub>, . . . , x<sub>n</sub>) and u<sub>n</sub> = h(x<sub>1</sub>, . . . , x<sub>n</sub>), is called a 100γ% [[confidence interval]] for θ. The number γ is called the [[confidence level]].<ref name=":0" /> In general, with a normally-distributed sample mean, Ẋ, and with a known value for the standard deviation, σ, a 100(1-α)% confidence interval for the true μ is formed by taking Ẋ ± e, with e = z<sub>1-α/2</sub>(σ/n<sup>1/2</sup>), where z<sub>1-α/2</sub> is the 100(1-α/2)% cumulative value of the standard normal curve, and n is the number of data values in that column. For example, z<sub>1-α/2</sub> equals 1.96 for 95% confidence.<ref>{{Cite book|title=Experimental Design – With Applications in Management, Engineering, and the Sciences|publisher=Paul D. Berger, Robert E. Maurer, Giovana B. Celli|year=2019|location=Springer}}</ref>

Here two limits are computed from the set of observations, say l<sub>n</sub> and u<sub>n</sub> and it is claimed with a certain degree of confidence (measured in probabilistic terms) that the true value of γ lies between l<sub>n</sub> and u<sub>n</sub>. Thus we get an interval (l<sub>n</sub> and u<sub>n</sub>) which we expect would include the true value of γ(θ). So this type of estimation is called confidence interval estimation.<ref name=":1" /> This estimation provides a range of values which the parameter is expected to lie. It generally gives more information than point estimates and are preferred when making inferences. In some way, we can say that point estimation is the opposite of interval estimation.