Probability theory: calculating confidence intervals

1. mean value 、 variance 、 Standard deviation

2. confidence interval

Confidence interval is a commonly used interval estimation method , So-called confidence interval That is, in terms of statistics Upper confidence limit and Lower confidence limit An interval consisting of upper and lower bounds .

• Significance level ：α
• Degree of confidence ：1-α perhaps 100（1-α）%  （ for example ,α=0.05, Then the confidence is 0.95 or 95%）
• Common calculation methods of confidence interval ：Pr（c1<=μ<=c2）=1-α
• confidence interval ：（c1, c2）

1） Total variance It is known that , The confidence interval of the population mean is ：

• That's the sample mean , That is, the arithmetic mean of all measured data .
• α It's the significance level ,α=1- Degree of confidence , For example, confidence is 95%, be α=1-0.95=0.05.
•   be called Z value , It can be obtained by looking up the normal distribution table .
• Is the standard deviation of the population ,n It's the number of samples .  It is called the standard error of the sample (standard error, SE).

2） The population variance is unknown , The confidence interval of the population mean is ：

• That's the sample mean , That is, the arithmetic mean of all measured data .
• α It's the significance level ,α=1- Degree of confidence , For example, confidence is 95%, be α=1-0.95=0.05.
• n It's the number of samples ,n-1 Become degrees of freedom , be called t value , You can check t The distribution table gives （ Yes ）.
• S That's the standard deviation of our sample ,n It's the number of samples .  It is called the average error of the sample .

The picture below is from a book 《 Probability theory and mathematical statistics 》：

Reference resources ：

 https://blog.csdn.net/alyssa520/article/details/85329083