ConfidenceIntervals.ppt

1、Confidence Intervals,置信区间,Copyright Route Six Sigma, LLC 2002,3,What Is a Confidence Interval,A confidence interval describes range of plausible values for a population parameter. Interval size is based on one of several statistical distributions. For large sample sizes (30) or when using population

2、 statistics, generally the Z distribution is used. For small samples (30), the student t distribution is commonly used. The boundary between the confidence interval and the area of risk is defined by the level of risk we are willing to assume, a. a is also called the significance level Confidence in

3、tervals can be defined with two boundaries or with a single boundary.,If the value falls in this region, we take a risk to say that it does not represent the true parameter measurement.,or,Copyright Route Six Sigma, LLC 2002,4,什么是置信区间,置信区间描述了总体参数可能的数值范围. 置信区间的大小是以统计数据分布状态为基础的。 当样本数量较多(30)或使用总体统计方法时，

4、通常采用Z分布 当样本数量较少(30)时,一般采用 student t分布. 我们愿意接受的风险水平a决定了置信区间和风险区域的分界线. a 也被称为显著水平 采用双边或单边界定置信区间均可.,若数值落于该区间，那么它可能并未反映真正的参数水平.,or,Copyright Route Six Sigma, LLC 2002,5,Example Two Sided Confidence Interval,95% of the samples taken from a population with a mean “m” can be observed within 1.96 of m. This

5、 is known as a 95% confidence interval. It is written Or The constant 1.96 is a Z Value taken from the Normal Distribution Table. Find 1.96. Note that it corresponds to 2.5E-02. Why .025? For this example, the confidence interval takes the 95% from the center of the distribution.,95%,2.5%,2.5%,Copyr

6、ight Route Six Sigma, LLC 2002,6,以双边置信区间为例,从总体中提取的95%样本的均值 “m” 被测出为 1.96 以內，称为95%的置信区间。该区间用公式表达如下： 或 常数 1.96 是取于正态分布表的Z值. 找到 1.96. 注意常数与 2.5E-02相对应. 为何是0 .025呢? 在此例中，置信区间在分布中心两端所占区域共计95%.,95%,2.5%,2.5%,Copyright Route Six Sigma, LLC 2002,7,Example One Sided Confidence Interval,If we are testing for

7、the mean being less than a number, well use a one sided test as shown below.,If we are testing for the mean being greater than a number, well use a one sided test as shown below.,In this case, with a=.05, the entire risk area is on one side. Write the inequality for each using the appropriate Z.,Cop

8、yright Route Six Sigma, LLC 2002,8,以单边置信区间为例,若测试的均值是小于一定数值的，我们将采用如下所示的单边测试方法.,若测试的均值是大于一定数值的，我们将采用如下所示的单边测试方法.,在此a=.05的例子中，整体风险区域是在一边的. 用适当的Z值写出相对应的不等式.,Copyright Route Six Sigma, LLC 2002,9,The Effect Of Level of Significance (a) On Confidence Interval,What happens when you change the desired signi

9、ficance level (a)?,As you increase the risk of claiming something is different when it is really the same (i.e., rejecting good parts at the assembly line), a gets larger and the confidence interval (1-a) grows smaller. As the you decrease the risk by decreasing a, the confidence interval (1-a) incr

10、eases.,Minitab will provide a p value with all confidence interval calculations. p value is the observed level of significance. If pa, There is insufficient evidence to support rejecting HO.,Copyright Route Six Sigma, LLC 2002,10,显著水平（a）对置信区间的影响,当合意的显著水平（ a ）被改变时会发生什么情况呢？,当增加把一致的零件视为不一的风险（如弃丟装配线中良好的

11、零件）时, 则a 值变大而置信区间(1-a)变小. 当通过减少a值来降低风险时，置信区间 (1-a) 就会增大.,Minitab 将为所有置信区间提供P值. p值是观测到的显著水平. 若 pa, 则沒有足够的证据支持弃用 HO.,Copyright Route Six Sigma, LLC 2002,11,The Effect of Sample Size (n) On Confidence Interval,What happens to the confidence interval as n decreases? The sigma multiple grows larger and t

12、he resultant Confidence interval increases. When the sample size (n) increases, the confidence interval decreases.,Copyright Route Six Sigma, LLC 2002,12,样本多少对置信区间（n）的影响,当n下降，置信区间会发生什么变化? Sigma 倍数变大，最终置信区间增大. 当样本数量 (n) 增加, 置信区间变小.,Copyright Route Six Sigma, LLC 2002,13,An Example,The nominal torque

13、of a finished awning tensioner is 200 foot lbs. A sample of 10 parts from one of the torsioning stations yielded an average torsion of 198.75lbs with an s=2.333 lbs. Are the parts being made to spec? Heres the test result,How about a 30 piece sample?,How does the answer change if we increase the sam

14、ple size to 20 with the same results?,Copyright Route Six Sigma, LLC 2002,14,实例,一个完工的布篷应张环的额定扭力是200 英尺/lbs. 从提取的10个部件样本显示：平均扭转度为198.75lbs与 s=2.333 lbs. 这些部件符合规格标准吗? 测试结果如下,30个样本呢?,若将样本大小增加到20个，结果如何呢?,Copyright Route Six Sigma, LLC 2002,15,Confidence Intervals Can Be Calculated for a Single Sample or

15、 for Two Samples,One Sample In a single sample analysis, the actual parameter measurement is the focus of the confidence interval. The confidence interval describes the range of plausible values of the parameter and that is compared to the target value (I.e., the nominal part size). The Null Hypothe

16、sis always includes the target in the confidence interval.,Two Samples In a two sample scenario, the difference in the parameter measurements is the focus. If the measurements are the same, the difference between them is zero and that is the target for the Confidence Interval. In such case, the valu

17、e zero (0) will always be included in the confidence interval.,Copyright Route Six Sigma, LLC 2002,16,置信区间可用于单个或两个样本计算,一个样本 在一个样本的分析中，确实地度量参数是置信区间的重点. 置信区间描述了与目标值相比的总体参数的可能数值范围 (如，理论上零件的大小). 虚无假设总是含括了置信区间的目标。,两个样本 在两个样本的情况下，参数被度量到的差异是焦点. 当测度方法一致时，两者之间的差异为零，而这也正是置信区间的目标所在. 在此例中，数值零(0)总是含括在置信区间中的.,Copyright Route Six Sigma, LLC 2002,17,Calculating the Z Value,We define the confidence interval as the range of plausible values of a statistic for a population parameter based on the selected value of a. We then compare the sampl