Statistics 简明教程

Statistics - Required Sample Size

检验的一个关键部分是选择检验测量值,即从人群中选择的要用于完成探索的单位数量。对于表征最合适的规模,没有明确的答案或答案。关于检验范围存在一些错误的判断,例如样本应该是人群的 10% 或样本规模与总体的大小有关。然而,如前所述,这些只是错误的判断。样本应有多大是所研究的人群参数中的差异容量,以及专家所需的评估准确性。

对最佳样本规模的决策可以从主观和数学两个角度进行。

Subjective Approach to Determining Sample Size

样本规模的选择受以下所讨论的各种因素影响:

  1. The Nature of Population - The level of homogeneity or heterogeneity influences the extent of a specimen. On the off chance that the populace is homogeneous concerning the qualities of interest then even a little size of the specimen is adequate. However in the event that the populace is heterogeneous then a bigger example would be required to guarantee sufficient representativeness.

  2. Nature of Respondent - If the respondents are effortlessly accessible and available then required data can be got from a little example. On the off chance that, notwithstanding, the respondents are uncooperative and non-reaction is relied upon to be high then a bigger specimen is required.

  3. Nature of Study - A onetime study can be led utilizing a substantial example. If there should be an occurrence of examination studies which are of constant nature and are to be seriously completed, a little specimen is more suitable as it is anything but difficult to oversee and hold a little example over a long compass of time.

  4. Sampling Technique Used - An essential variable affecting the span of test is the examining system received. Firstly a non-likelihood system requires a bigger specimen than a likelihood strategy. Besides inside of likelihood testing, if straightforward irregular examining is utilized it requires a bigger example than if stratification is utilized, where a little specimen is adequate.

  5. Complexity of Tabulation - While settling on the specimen estimate the specialist ought to likewise consider the quantity of classifications and classes into which the discoveries are to be assembled and broke down. It has been seen that more the quantity of classifications that are to be produced the bigger is the example size. Since every class ought to be enough spoken to, a bigger specimen is required to give solid measures of the littlest classification.

  6. Availability of Resources - The assets and the time accessible to specialist impact the span of test. Examination is a period and cash escalated assignment, with exercises like readiness of instrument, contracting and preparing field staff, transportation costs and so forth taking up a considerable measure of assets. Subsequently if the scientist does not have enough time and supports accessible he will settle on a littler example.

  7. Degree of Precision and Accuracy Required - . It has turned out to be clear from our prior discourse that accuracy, which is measured by standard blunder, wills high just if S.E is less or the example size is substantial.

Also to get a high level of precision a bigger specimen is required. Other then these subjective efforts, sample size can be determined mathematically also.

Mathematical Approach to Sample Size Determination

In the mathematical approach to sample size determination the precision of estimate required is stated first and then the sample size is worked out. The precision can be specified as ${\pm}$ 1 of the true mean with 99% confidence level. This means that if the sample mean is 200, then the true value of the mean will be between 199 and 201. This level of precision is denoted by the term 'c'

Sample Size determination for means.

The confidence interval for the universe mean is given by

其中——

  1. ${\bar x}$ = Sample mean

  2. ${e}$ = Acceptable error

  3. ${Z}$ = Value of standard normal variate at a given confidence level

  4. ${\sigma_p}$ = Standard deviation of the population

  5. ${n}$ = Size of the sample

The acceptable error 'e' i.e. the difference between ${\mu}$ and ${\bar x}$ is given by

Thus, Size of the sample is:

In case the sample size is significant visa-a-vis the population size then above formula will be corrected by the finite population multiplier.

其中——

  1. ${N}$ = size of the population

Sample Size Determination for Proportions

The method for determining the sample size when estimating a proportion remains the same as the method for estimating the mean. The confidence interval for universe proportion ${\hat p}$ is given by

其中——

  1. ${p}$ = sample proportion

  2. ${q = (1 - p)}$

  3. ${Z}$ = 样本比例的标准正态变异值

  4. ${n}$ = Size of the sample

由于 ${ \hat p}$ 将被估计,因此可以通过采用一个可接受值 p = 0.5 来确定 p 的值,进而得出保守样本量。另一种选择是通过试点研究或根据个人判断来估计 p 的值。已知 p 的值,则可接受的误差“e”表示为:

如果总体是有限的,那么上述公式将由有限总体乘数进行修正。

Example

Problem Statement:

一家商店有兴趣估计拥有商店特权会员卡的家庭的比例。以前的研究表明,59% 的家庭拥有商店信用卡。在 95% 的置信度下,容忍误差水平为 05。

Solution:

该商店拥有以下信息:

可以通过应用以下公式来确定样本量:

因此,369 户家庭的样本足以进行研究。

由于人口,即目标家庭被认为是 1000 户,并且上述样本是总人口的很大一部分,因此使用了包含有限总体乘数的修正公式。

因此,如果人口是 1000 户的有限人口,那么进行研究所需的样本量是 270 户。

从这个例子中可以明显看出,如果已知总体规模,则确定的样本量就会减少。