Statistics 简明教程

Statistics - Interval Estimation

区间估计是利用样本数据计算未知总体参数的可能（或可概）值区间，这与点估计不同，后者只有一个数字。

Interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter, in contrast to point estimation, which is a single number.

Formula

其中——

Where −

  1. ${\bar x}$ = mean

  2. ${Z_{\frac{\alpha}{2}}}$ = the confidence coefficient

  3. ${\alpha}$ = confidence level

  4. ${\sigma}$ = standard deviation

  5. ${n}$ = sample size

Example

Problem Statement:

Problem Statement:

假设某个学生测量某一液体的沸点并得到 6 个不同液体的读数（以摄氏度为单位）：102.5、101.7、103.1、100.9、100.5 和 102.2。他计算出样本均值为 101.82。如果他知道此过程的标准偏差为 1.2 度，他在 95% 置信水平下对总体均值进行区间估计是什么？

Suppose a student measuring the boiling temperature of a certain liquid observes the readings (in degrees Celsius) 102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different samples of the liquid. He calculates the sample mean to be 101.82. If he knows that the standard deviation for this procedure is 1.2 degrees, what is the interval estimation for the population mean at a 95% confidence level?

Solution:

Solution:

该学生计算出沸点的样本均值为 101.82，标准偏差为 ${\sigma=0.49}$。95% 置信区间的临界值为 1.96，其中 ${\frac{1-0.95}{2}=0.025}$。未知均值的 95% 置信区间。

The student calculated the sample mean of the boiling temperatures to be 101.82, with standard deviation ${\sigma = 0.49}$. The critical value for a 95% confidence interval is 1.96, where ${\frac{1-0.95}{2} = 0.025}$. A 95% confidence interval for the unknown mean.

随着置信水平的降低，相应区间的长度将会减小。假设该学生对沸点的 90% 置信区间感兴趣。在这种情况下，${\sigma=0.90}$，${\frac{1-0.90}{2}=0.05}$。此水平的临界值等于 1.645，因此 90% 置信区间为

As the level of confidence decreases, the size of the corresponding interval will decrease. Suppose the student was interested in a 90% confidence interval for the boiling temperature. In this case, ${\sigma = 0.90}$, and ${\frac{1-0.90}{2} = 0.05}$. The critical value for this level is equal to 1.645, so the 90% confidence interval is

增加样本量将减少置信区间长度，而不会降低置信水平。这是因为标准偏差随着 n 的增加而减小。

An increase in sample size will decrease the length of the confidence interval without reducing the level of confidence. This is because the standard deviation decreases as n increases.

Margin of Error

区间估计的误差范围 ${m}$ 定义为从样本均值中添加或减去的确定区间的长度的值：

The margin of error ${m}$ of interval estimation is defined to be the value added or subtracted from the sample mean which determines the length of the interval:

假设在上面的示例中，该学生希望在 95% 置信度的情况下具有 0.5 的误差范围。将适当的值代入 ${m}$ 的表达式并求解 n 即可得到计算结果。

Suppose in the example above, the student wishes to have a margin of error equal to 0.5 with 95% confidence. Substituting the appropriate values into the expression for ${m}$ and solving for n gives the calculation.

为了对总长度小于 1 度的平均沸点实现 95% 区间估计，该学生将不得不进行 23 次测量。

To achieve 95% interval estimation for the mean boiling point with total length less than 1 degree, the student will have to take 23 measurements.