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To determine an appropriate sample size n for estimating proportions, the equation below can be solved, where W represents the desired width of the confidence interval. The resulting sample size formula, is often applied with a conservative estimate of p (e.g., 0.5): = /
Formulas, tables, and power function charts are well known approaches to determine sample size. Steps for using sample size tables: Postulate the effect size of interest, α, and β. Check sample size table. Select the table corresponding to the selected α; Locate the row corresponding to the desired power; Locate the column corresponding to ...
This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike. See also unbiased estimation of standard deviation for more discussion.
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model , the observed data is most probable.
If the sampling design is one that results in a fixed sample size n (such as in pps sampling), then the variance of this estimator is: Var ( Y ¯ ^ known N ) = 1 N 2 n n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 {\displaystyle \operatorname {Var} \left({\hat {\bar {Y}}}_{{\text{known }}N}\right)={\frac {1}{N^{2}}}{\frac {n}{n-1}}\sum _{i=1 ...
where m is the sample maximum and k is the sample size, sampling without replacement. This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II. The formula may be understood intuitively as;
where n is the sample size and N is the population size and s xy is the covariance of x and y. An estimate accurate to O( n −2 ) is [3] var ( r ) = 1 n [ s y 2 m x 2 + m y 2 s x 2 m x 4 − 2 m y s x y m x 3 ] {\displaystyle \operatorname {var} (r)={\frac {1}{n}}\left[{\frac {s_{y}^{2}}{m_{x}^{2}}}+{\frac {m_{y}^{2}s_{x}^{2}}{m_{x}^{4 ...
Thus the estimation of covariance matrices directly from observational data plays two roles: to provide initial estimates that can be used to study the inter-relationships; to provide sample estimates that can be used for model checking.
The minimum and the maximum value are the first and last order statistics (often denoted X (1) and X (n) respectively, for a sample size of n). If the sample has outliers , they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low.
To estimate the variance σ 2, one estimator that is sometimes used is the maximum likelihood estimator of the variance of a normal distribution σ ^ 2 = 1 n ∑ ( X i − X ¯ ) 2 . {\displaystyle {\widehat {\sigma }}^{2}={\frac {1}{n}}\sum \left(X_{i}-{\overline {X}}\right)^{2}.}