|
|
| แถว 1: |
แถว 1: |
| − | == Sample == | + | == Sampling and statistics == |
| − | Consider a certain distribution. The mean <math>\mu</math> of the distribution is the expected value of a random variable <math>X</math> sample from the distribution. I.e.,
| |
| | | | |
| − | <center>
| + | == Confidence interval, known <math>\sigma</math> == |
| − | <math>\mu=E[X]</math>. | |
| − | </center>
| |
| | | | |
| − | Also recall that the variance of the distribution is
| + | == Confidence interval, unknown <math>\sigma</math> == |
| − | | |
| − | <center>
| |
| − | <math>\sigma^2=Var(X)=E[(X-\mu)^2]=E[X^2] = E[X]^2.</math>.
| |
| − | </center>
| |
| − | | |
| − | And finally, the standard deviation is <math>\sigma = \sqrt{Var(X)}</math>.
| |
| − | | |
| − | Suppose that you take <math>n</math> samples <math>X_1,X_2,\ldots,X_n</math> independently from this distribution. (Note that <math>X_1,X_2,\ldots,X_n</math> are random variables.
| |
| − | | |
| − | === Sample means ===
| |
| − | | |
| − | The statistic
| |
| − | | |
| − | <center>
| |
| − | <math>\bar{X} = \frac{X_1+X_2+\cdots+X_n}{n} = \frac{1}{n}\sum_{i=1}^n X_i</math>
| |
| − | </center>
| |
| − | | |
| − | is called a '''sample mean.''' Since <math>X_1,X_2,\dots,X_n</math> are random variables, the mean <math>\bar{X}</math> is also a random variable.
| |
| − | | |
| − | Thus, we can compute:
| |
| − | | |
| − | <math>E[\bar{X}]= E\left[\frac{1}{n}\sum_{i=1}^n X_i\right] = \frac{1}{n}E\left[\sum_{i=1}^n X_i\right] = \frac{1}{n}\sum_{i=1}^n E[X_i] = \frac{1}{n} n\mu = \mu</math>
| |
| − | | |
| − | and
| |
| − | | |
| − | <math>Var(\bar{X}) = \frac{\sigma^2}{n}.</math>
| |
| − | | |
| − | === Sample variances and sample standard deviations ===
| |
