ผลต่างระหว่างรุ่นของ "Probstat/notes/confidence intervals"

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: ''This is part of [[probstat]]''
 
: ''This is part of [[probstat]]''
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Suppose that we take a sample of size <math>n</math>, <math>X_1,X_2,\ldots,X_n</math> from a population which is normally distributed.  Also suppose that the population has mean <math>\mu</math> and variance <math>\sigma^2</math>.  In this section, we assume that we do not know <math>\mu</math> but we know the variance <math>\sigma^2</math>.  The case we the variance is unknown will be discussed [[Probstat/notes/t-distributions|here]].
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We would like to estimate the mean.  Therefore, we compute the sample mean <math>\bar{X}</math>.    It is very certain the <math>\bar{X}\neq\mu</math>, but we hope that it will be close to <math>\mu</math>.  In this section, we try to quantify how close the sample mean to the real mean.
  
 
Recall that the random variable <math>\bar{X}</math> is a normal random variable with mean <math>\mu</math> and s.d. <math>\sigma/\sqrt{n}</math>  Therefore,
 
Recall that the random variable <math>\bar{X}</math> is a normal random variable with mean <math>\mu</math> and s.d. <math>\sigma/\sqrt{n}</math>  Therefore,

รุ่นแก้ไขเมื่อ 15:49, 4 ธันวาคม 2557

This is part of probstat

Suppose that we take a sample of size , from a population which is normally distributed. Also suppose that the population has mean and variance . In this section, we assume that we do not know but we know the variance . The case we the variance is unknown will be discussed here.

We would like to estimate the mean. Therefore, we compute the sample mean . It is very certain the , but we hope that it will be close to . In this section, we try to quantify how close the sample mean to the real mean.

Recall that the random variable is a normal random variable with mean and s.d. Therefore,

will be a unit normal random variable.

We consider how deviates from the true mean .

To be added