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代写coursework,Analysis of Data
发表日期:2013-10-11 08:51:38 | 来源:assignment.cc | 当前的位置:首页 > 代写coursework > 正文
Consider and discuss the required approach to fullanalysis of the data set provided.
As part of this explore also how you would test the hypothesis below and explain the reasons for your decisions. Hypothesis 1: Male children are taller than female children. Null hypothesis; There is no difference in height between male children and female children. Hypothesis 2: Taller children are heavier. Null hypothesis: There is no relationship between how tall children are and how much they weigh.
Analysis of data set
The data set is a list of 30 children's gender, age, height,weight, upper and lower limb lengths, eye colour, like of chocolate or not andIQ.
There are two main things to consider before analysing thedata. These are the types of data and the quality of the data as a sample.
Types of data could be nominal, ordinal, interval or ratio.Nominal is also know as categorical. Coolican (1990) gives more details of allof these and his definitions have been used to decide the types of data in thedata set.
It is also helpful to distinguish between continuousnumbers, which could be measured to any number of decimal places an discretenumbers such as integers which have finite jumps like 1,2 etc.
Gender
This variable can only distinguish between male or female.There is no order to this and so the data is nominal.
Age
This variable can take integer values. It could be measuredto decimal places, but is generally only recorded as integer. It is ratio databecause, for example, it would be meaningful to say that a 20 year old personis twice as old as a 10 year old.
In this data set, the ages range from 120 months to 156months. This needs to be consistent with the population being tested.
Height
This variable can take values to decimal places ifnecessary. Again it is ratio data because, for example, it would be meaningfulto say that a person who is 180 cm tall is 1.5 times as tall as someone 120cmtall. In this sample it is measured to the nearest cm.
Weight
Like height, this variable could take be measured to decimalplaces and is ratio data. In this sample it is measured to the nearest kg.
Upper and lower limb lengths
Again this variable is like height and weight and is ratiodata.
Eye colour
This variable can take a limited number of values which areeye colours. The order is not meaningful. This data is therefore nominal(categorical).
Like of chocolate or not
As with eye colour, this variable can take a limited numberof values which are the sample members preferences. In distinguishing merelybetween liking and disliking, the order is not meaningful. This data istherefore nominal (categorical).
IQ
IQ is a scale measurement found by testing each samplemember. As such it is not a ratio scale because it would not be meaningful tosay, for example, that someone with a score of 125 is 25% more intelligent thansomeone with a score of 100.
There is another level of data mentioned by Cooligan intowhich none of the data set variables fit. That is Ordinal Data. This means thatthe data have an order or rank which makes sense. An example would be if 10students tried a test and you recorded who finished quickest, 2ndquickest etc, but not the actual time.
The data is intended to be a sample from a population aboutwhich we can make inferences. For example in the hypothesis tests we want toknow whether they are indicative of population differences. The results canonly be inferred on the population from which it is drawn it would not be validotherwise.
Details of sampling methods were found in Bland (2000). Toaccomplish the required objectives, the sample has to be representative of thedefined population. It would also be more accurate if the sample is stratifiedby known factors like gender and age. This means that, for example, theproportion of males in the sample is the same as the proportion in thepopulation.
Sample size is another consideration. In this case it is 30.Whether this is adequate for the hypotheses being tested is examined below.
Hypothesis 1: Male children are taller than femalechildren.
Swift (2001) gives a very readable account of the hypothesistesting process and the structure of the test.
The first step is to set up the hypotheses:
The Null hypothesis is that there is no difference in heightbetween male children and female children.
If the alternative was as Coolican describes it as "wedo not predict in which direction the results will go then it would have beena two-tailed test. In this case the alternative is that males are taller it istherefore a specific direction and so a one-tailed test is required.
To test the hypothesis we need to set up a test statisticand then either match it against a pre-determined critical value or calculatethe probability of achieving the sample value based on the assumption that thenull hypothesis is true.
The most commonly used significance level is 0.05. Accordingto Swift (2001) the significance level must be decided before the data isknown. This is to stop researchers adjusting the significance level to get theresult that they want rather than accepting or rejecting objectively.
If the test statistic probability is less than 0.05 we wouldreject the null hypothesis that there is no difference between males andfemales in favour of males being heavier on the one sided basis.
However it is possible for the test statistic to be in therejection zone when in fact the null hypothesis is true. This is called a TypeI error.
It is also possible for the test statistic to be in theacceptance zone when the alternative hypothesis is true (in other words thenull hypothesis is false). This is called a Type II error. Power is 1 -probability of a Type II error and is therefore the probability of correctlyrejecting a false null hypothesis. Whereas the Type I error is set at thedesired level, the Type II error depends on the actual value of the alternativehypothesis.
Coolican (1990) sets out the possible outcomes in thefollowing table:
  In acceptance zone In rejection zone
NULL Hypothesis TRUE OK Type I error
NULL Hypothesis FALSE Type II error OK
Test method
The data for gender is categorical and for height the datais ratio. The sample is effectively split into 2 sub-sets for male and female.
Most books give the independent samples t-test as the mainmethod for testing this hypothesis e.g. Curwin, et al (2001), Swift (2001).
Bland (2000) states that in order to use this test thesamples must both be from a normal populations and additionally thedistributions must have the same variance. Bland also suggests modifications tothe test when the variances cannot be assumed to be the same. Programs likeSPSS will calculate both for equal and non-equal variances. SPSS also gives atest for equality of variances.
When the assumptions of normality and independence are metthen the t-test is the best test according to Bland because it has higher powerthan the equivalent non-parametric test which is the Mann-Whitney U-test.However, the Mann-Whitney test is more robust in that it does not assume thatthe data is normally distributed.
It is a matter of weighing up the pros and cons. Ifnormality can be assumed then the independent samples t-test is best. If not,then the U-test should be used. Tests suchj as a histogram or Q-Q(quantile-quantile) plot can be used to check normality to help the decisionBland (2000).
Because the test is one-sided we would be looking for themale mean to be higher and the critical value to come from 0.05 in the one tailof the distribution. For the t test this would be looked up with n1 + n2 - 2degrees of freedom where n1 and n2 are the numbers of males and femalesrespectively.
It is also useful to work out a 95% confidence interval forthe population mean. This gives an idea of the spread of the estimate. Largersample sizes will reduce the confidence interval.
It was mentioned above that the inferences made are onlyvalid for the population being sampled and only so if the sample isrepresentative, which means selecting the sample from the whole population suchthat each member has equal probability of selection.
For the results to be reliable as Coolican (1990) says thatif a research finding can be repeated it is reliable. So, if the sample isrepeated the same result would indicate reliability.
Hypothesis 2: Taller children are heavier.
The null hypothesis is that there is no relationship betweenhow tall children are and how much they weigh. The alternative hypothesis isthat taller children are heavier, which is a one-sided test. That is, thealternative is not simply that there is a relationship, which would betwo-sided.
Both heights and weights are ratio data. This enables thedata to be examined by tests where normality is an underlying assumption.
In order to visually check the relationship a scatter graphis pretty well essential. This would give an idea of the strength and nature ofthe relationship. The relationship may not be linear as is often assumed. If sothen the scatter should show indication of a curve.
The strength of the relationship can be tested by using thePearson correlation coefficient ( r ). This is closely related to a regressionanalysis which would be fitting a straight line equation to the data withheight being the independent (x) variable and weight being dependent (y).
The correlation coefficient can be tested using a 1 sidedt-test. This has n-2 degrees of freedom, 28 in this case. The value of r wouldneed to be positive to indicate that taller children are heavier.
Analysis of the regression residuals can give us a lot moreinformation than simply carrying out a correlation calculation. See Bland(2000). They can be plotted to see whether they are normally distributed usinga histogram or Q-Q plot. Also, non-linearity should be apparent if this is thecase.
If the data shows a non-linear relationship then it would benecessary to transform the data using logs or other mathematical functions. Thetransformed variables would then need to be analyses for normality andlinearity.
According to Bland there is an alternative to the Pearsoncorrelation coefficient which does not assume that the data is normallydistributed. This is the Spearman Rank Correlation Coefficient. This is basedon the distribution of the ranks of the data and not the data itself. Thismakes it more robust in terms of departures from assumptions, however it isless powerful. In other words there is more chance of making a Type II error.
Again, if the sample is repeated the same result wouldindicate reliability.
Summary
The stages that need to be gone through in order to testhypotheses such as those above is as follows.
  • Definethe population about which inferences are to be made. This acts as a basis fromwhich to obtain a sample. The results will only apply to this population.
  • Decidehow to obtain a representative sample from the population. Decide the sample sizerequired and whether the sample can be stratified to make it more accurate.There is a large amount of literature available to help with sampling.
  • Setup null and alternative hypotheses giving particular attention to whether thealternative should be one or two sided.
  • Decideon the significance level before calculating the test statistic. This isusually 0.05 but sometimes 0.01. To be objective the value must be set beforecarrying out the analysis.
  • Choosethe testing method. Particular emphasis should be placed on the assumptionsthat the rest requires, particularly the assumption of normality which isneeded for tests like t tests. The pros and cons of the various alternativesshould be weighed up. This often boils down to power versus robustness.
  • Checknormality and linearity where appropriate by drawing graphs like QQ andhistograms for normality and scatter graphs for exploring relationships.
  • Calculatethe test statistic and compare with the critical value. Also calculate theprobability of obtaining the sample value. Reject the null hypothesis if thesample value is outside the critical range or single value if it is a one-sidedtest.
  • Beaware of the possibility of a Type II error which is accepting the nullhypothesis when it is in fact false.
References
Bland, J.Martin (2000) An Introduction to MedicalStatistics 3rd Edition Oxford. Oxford Medical Publications.
Curwin, Jon and Slater, Roger (2001) QuantitativeMethods for Business Decisions
London, Thomson Learning
Coolican, Hugh (1990) Research Methods and Statistics inPsychology London, Hodder and Stoughton
Swift, Louise (2001) Quantitative Methods for Business,Management and Finance, Basingstoke, Palgrave
 
审议和讨论所需的做法到fullanalysis的数据集提供。
作为这项工作的一部分,也探讨如何将测试下面的假设和解释的理由为自己的决定。假设1:男的孩子都长得比女性儿童。零假设有没有男性儿童和女性儿童之间的高度差。假设2:身材较高的子女也较重。零假设:有没有关系如何高大的孩子们,他们的体重多少。
分析数据集
该数据集是30个孩子的性别,年龄,身高,体重,上下肢的长度,眼睛的颜色,喜欢的巧克力或不andIQ的列表。
有两种主要的事情要考虑,分析捜之前。这些类型的数据和作为试样的数据的质量。
类型的数据可以是名义,有序,间隔或ratio.Nominal的分类也知道。 coolican (1990)给出了这些allof运算的更多的细节,他的定义已被用来决定海图集的数据的类型。
这也有利于区分,可以测量到任意数量的小数位数有限的跳跃,如1,2等整数,如一个discretenumbers continuousnumbers
性别
这个变量只能分辨男性或female.There之间没有为了这一点,所以数据是名义。
年龄
这一变量可以取整数值。这可能是measuredto小数,但一般只记录为整数。例如,它是比databecause ,它会是有意义的说,一个20岁的personis两次作为一个10岁的老。
在这组数据中,年龄范围从120个月156个月。这与被检测人群的需要是一致的。
高度
这个变量的值可以到小数点后ifnecessary 。这又是比数据,例如,因为这将是meaningfulto说,一个人谁是180厘米高是有人120cmtall高1.5倍。在此示例中,它是计算至最接近的cm 。
重量
喜欢的高度,这个变量可以被测量的小数位数比数据。在此示例中,它测量到最近的公斤。
上肢和下肢的长度
同样这个变量,如身高和体重,并是ratiodata 。
眼睛颜色
这一变量可以取的值为areeye颜色数目有限的。顺序没有意义。因此,这个数据是名义(分类) 。
喜欢的巧克力或不
与眼睛的颜色一样,可以采取这个变量的值数量有限的样本成员的喜好。在区分merelybetween喜欢和不喜欢中,顺序是没有意义的。此数据istherefore标称(分类) 。
智商
IQ是一个尺度测量通过测试每个samplemember发现。因此,它是不是比规模,因为它不会有意义tosay的,例如,有人得分125 , 100的得分是25%以上的智能thansomeone 。
还有另外一个级别的数据集变量拟合Cooligan intowhich没有提到的数据。这是有序数据。这意味着thatthe数据有一个订单或等级,这是有道理的。一个例子是,如果10students试图测试完成最快, 2ndquickest等,但不实际的时间记录。
数据乃拟从人口aboutwhich的是一个示例,我们可以作出推论。例如,在假设检验,我们希望无论他们是toknow人口差异的指标。结果canonly推断的人口从它是它不会是validotherwise绘制。
布兰德( 2000年)被发现在抽样方法详情。 Toaccomplish所要求的目标,样品有代表人口thedefined 。这也将是更准确的,如果采样stratifiedby已知的性别和年龄等因素。这意味着,例如,样品中的男性theproportion作为在thepopulation的比例是相同的。
样本大小是另一个考虑因素。在这种情况下,这是足够的,对于被测试的假说如下检查30.Whether 。
假设1:男性儿童长得比femalechildren 。
斯威夫特(2001)给出了一个非常可读的帐户的hypothesistesting过程和结构的测试。
第一步是成立的假设:
零假设是,在heightbetween男性儿童和女性儿童没有任何区别。
如果选择为Coolican把它描述为“我们做不预测结果将在哪个方向去,那么这将有beena双尾检验,在这种情况下,另一种方法是,男性有更高的它istherefore的一个特定的方向,所以一个尾检验是必需的。
为了检验这一假设,我们需要建立一个测试statisticand然后要么匹配它预先设定的临界值或calculatethe的概率实现采样值的基础上的假设,假设是正确的thenull 。
最常用的显着性水平为0.05 。根据斯威夫特(2001)显着性水平必须决定数据之前isknown 。这是停止调整的显着性水平的研究人员获得theresult他们想要的,而不是客观地接受或拒绝。
如果检验统计量的概率是小于0.05 ,我们wouldreject的零假设,没有任何区别男性片面的基础上重之间的男性andfemales的青睐。
然而,它有可能,测试统计在therejection区在事实上的零假设是真实的。这被称为一个TYPEI错误。
另外,也可以检验统计量时,在theacceptance区替代的假说是真实的(换句话说thenull假设为假) 。这就是所谓的第二类错误。功率为1一种Ⅱ型错误的概率,并因此correctlyrejecting虚假的零假设的概率。鉴于I型错误设置在thedesired的水平,第二类错误取决于实际值的alternativehypothesis 。
Coolican ( 1990年)载有可能的结果在下面的表:
在接受区
在排斥区
零假设为TRUE
I型错误
零假设为FALSE
第二类错误
测试方法
性别的数据分类和高度的datais比。样品被有效地分成2个子集,男性和女性。
大多数书籍给人的独立样本t检验测试这一假说,例如mainmethod Curwin等人(2001) ,斯威夫特(2001年) 。
布兰德(2000)指出,为了使用这种测试thesamples ,都必须从一个正常的人口和,附加thedistributions必须具有相同的方差。布兰德还建议修改tothe测试,不能假定方差是相同的。程序likeSPSS的都将计算为平等和非平等的差异。 SPSS也给测试平等的差异。
当正常和独立的假设metthen t检验是最好的测试根据布兰德,因为它具有较高的powerthan是采用Mann-Whitney U- test.However相当于非参数检验, Mann-Whitney检验强大,因为它不承担thatthe数据是正态分布的。
这是一个权衡的利弊问题。 Ifnormality可以假设,然后独立样本t检验是最好的。如果没有,那么U-检验应该被使用。测试suchj直方图或QQ (位数 - 分位数)的情节,可用于正常性检查,以帮助的decisionBland (2000) 。
由于测试是片面的我们将寻找themale ,意味着更高的临界值0.05在一个tailof分布。对于t检验,这将抬头N1 + N2 - 2degrees公司的自由,其中n1和n2的男性和femalesrespectively的是数字。
它也是有用的工作, 95 %的置信区间满心欢喜人口平均。这给出了一个理想的传播估计。 Largersample规模将减少的置信区间。
上述所作的推论onlyvalid被采样的人口只有如此,如果样品isrepresentative ,这意味着选择样本的整个人口suchthat的每个成员都有相同的概率选择。
结果是可靠的为Coolican (1990)说,它是可靠的,可重复的研究发现thatif 。所以,如果样品isrepeated相同的结果将表明其可靠性。
假设2:身材较高的子女也较重。
零假设是,有没有关系betweenhow高大的孩子们和他们的体重。替代假设isthat的身材较高的子女重,这是一种片面的测试。即, thealternative是不能简单地认为是有关系的, betwo双面。
身高和体重比数据。这使海图测试常态一个潜在的假设进行审查。
为了直观检查的分散GRAPHIS相当不错重要的关系。这将使的强度和性质的OFTHE关系的一个想法。这种关系可能不是线性的,如通常假定。如果sothen分散应该显示曲线的迹象。
强度的关系可以测试使用thePearson相关系数(r) 。这是密切相关的一个regressionanalysis将拟合直线方程是独立的(x)的变量和权重取决于数据withheight (y)的。
使用1 sidedt测试的可测试的相关系数。这具有在这种情况下,自由, 28度的n-2 。 r的值wouldneed是正的,表明身高的儿童重。
回归残差的分析可以给我们带来很多更多的信息比单纯进行相关计算。布兰德(2000年) 。他们可以绘制,看他们是否是正态分布usinga直方图或QQ图。另外,非线性应该是显而易见的,如果这是thecase 。
如果数据显示一个非线性的关系,那么这将benecessary使用日志或其他数学函数来转换数据。 Thetransformed变量,然后将需要要分析常态andlinearity的。
据布兰德还存在另一种的Pearsoncorrelation系数不承担数据normallydistributed的。这是Spearman等级相关系数。这是基于数据分布的行列,而不是数据本身。假设偏离Thismakes更强大的,但它isless强大。换言之,有更多的机会使II型错误。
再次,如果重复样品的相同的结果wouldindicate的可靠性。
总结
需要经历,以如上述的testhypotheses的阶段,如下所述。
Definethe人口作出推论。这样的行为作为基础fromwhich中,得到样品。结果将只适用于这一人群。
从人口Decidehow获得具有代表性的样本。决定样品sizerequired和样品是否可以分层,使其更多accurate.There是大量的文献可以帮助采样。
设置空和替代假说给予特别注意是否thealternative应该是一个或两个片面。
Decideon显着性水平,然后再计算检验统计量。这个isusually的0.05 ,但有时是0.01 。要客观的价值必须设置了分析beforecarrying 。
Choosethe测试方法。应特别强调放置在assumptionsthat上,其余的需要,尤其是正态性假设t检验,测试,如isneeded 。的利弊权衡各种alternativesshould的。这往往归结为功率与鲁棒性。
在适当的情况下绘制图形像QQ andhistograms的探索关系的常态和散点图Checknormality和线性度。
Calculatethe检验统计量和临界值进行比较。也计算theprobability取得的样本值。拒绝虚无假设如果thesample值超出临界范围或单值,如果它是一个sidedtest 。
Beaware接受nullhypothesis时,它其实也是假的II型错误的可能性。
参考文献
布兰德, J.Martin (2000)介绍到MedicalStatistics第三版牛津。牛津大学医学刊物。
Curwin ,乔恩和Slater ,罗杰(2001) QuantitativeMethods的商业决策
伦敦,汤姆森学习出版集团
休Coolican (1990)研究方法和统计inPsychology伦敦,霍德和斯托顿
斯威夫特,路易斯(2001)的定量方法商业,管理及财务,贝辛斯托克,帕尔格雷夫Consider and discuss the required approach to fullanalysis of the data set provided.
As part of this explore also how you would test the hypothesis below and explain the reasons for your decisions. Hypothesis 1: Male children are taller than female children. Null hypothesis; There is no difference in height between male children and female children. Hypothesis 2: Taller children are heavier. Null hypothesis: There is no relationship between how tall children are and how much they weigh.
Analysis of data set
The data set is a list of 30 children's gender, age, height,weight, upper and lower limb lengths, eye colour, like of chocolate or not andIQ.
There are two main things to consider before analysing thedata. These are the types of data and the quality of the data as a sample.
Types of data could be nominal, ordinal, interval or ratio.Nominal is also know as categorical. Coolican (1990) gives more details of allof these and his definitions have been used to decide the types of data in thedata set.
It is also helpful to distinguish between continuousnumbers, which could be measured to any number of decimal places an discretenumbers such as integers which have finite jumps like 1,2 etc.
Gender
This variable can only distinguish between male or female.There is no order to this and so the data is nominal.
Age
This variable can take integer values. It could be measuredto decimal places, but is generally only recorded as integer. It is ratio databecause, for example, it would be meaningful to say that a 20 year old personis twice as old as a 10 year old.
In this data set, the ages range from 120 months to 156months. This needs to be consistent with the population being tested.
Height
This variable can take values to decimal places ifnecessary. Again it is ratio data because, for example, it would be meaningfulto say that a person who is 180 cm tall is 1.5 times as tall as someone 120cmtall. In this sample it is measured to the nearest cm.
Weight
Like height, this variable could take be measured to decimalplaces and is ratio data. In this sample it is measured to the nearest kg.
Upper and lower limb lengths
Again this variable is like height and weight and is ratiodata.
Eye colour
This variable can take a limited number of values which areeye colours. The order is not meaningful. This data is therefore nominal(categorical).
Like of chocolate or not
As with eye colour, this variable can take a limited numberof values which are the sample members preferences. In distinguishing merelybetween liking and disliking, the order is not meaningful. This data istherefore nominal (categorical).
IQ
IQ is a scale measurement found by testing each samplemember. As such it is not a ratio scale because it would not be meaningful tosay, for example, that someone with a score of 125 is 25% more intelligent thansomeone with a score of 100.
There is another level of data mentioned by Cooligan intowhich none of the data set variables fit. That is Ordinal Data. This means thatthe data have an order or rank which makes sense. An example would be if 10students tried a test and you recorded who finished quickest, 2ndquickest etc, but not the actual time.
The data is intended to be a sample from a population aboutwhich we can make inferences. For example in the hypothesis tests we want toknow whether they are indicative of population differences. The results canonly be inferred on the population from which it is drawn it would not be validotherwise.
Details of sampling methods were found in Bland (2000). Toaccomplish the required objectives, the sample has to be representative of thedefined population. It would also be more accurate if the sample is stratifiedby known factors like gender and age. This means that, for example, theproportion of males in the sample is the same as the proportion in thepopulation.
Sample size is another consideration. In this case it is 30.Whether this is adequate for the hypotheses being tested is examined below.
Hypothesis 1: Male children are taller than femalechildren.
Swift (2001) gives a very readable account of the hypothesistesting process and the structure of the test.
The first step is to set up the hypotheses:
The Null hypothesis is that there is no difference in heightbetween male children and female children.
If the alternative was as Coolican describes it as "wedo not predict in which direction the results will go then it would have beena two-tailed test. In this case the alternative is that males are taller it istherefore a specific direction and so a one-tailed test is required.
To test the hypothesis we need to set up a test statisticand then either match it against a pre-determined critical value or calculatethe probability of achieving the sample value based on the assumption that thenull hypothesis is true.
The most commonly used significance level is 0.05. Accordingto Swift (2001) the significance level must be decided before the data isknown. This is to stop researchers adjusting the significance level to get theresult that they want rather than accepting or rejecting objectively.
If the test statistic probability is less than 0.05 we wouldreject the null hypothesis that there is no difference between males andfemales in favour of males being heavier on the one sided basis.
However it is possible for the test statistic to be in therejection zone when in fact the null hypothesis is true. This is called a TypeI error.
It is also possible for the test statistic to be in theacceptance zone when the alternative hypothesis is true (in other words thenull hypothesis is false). This is called a Type II error. Power is 1 -probability of a Type II error and is therefore the probability of correctlyrejecting a false null hypothesis. Whereas the Type I error is set at thedesired level, the Type II error depends on the actual value of the alternativehypothesis.
Coolican (1990) sets out the possible outcomes in thefollowing table:
  In acceptance zone In rejection zone
NULL Hypothesis TRUE OK Type I error
NULL Hypothesis FALSE Type II error OK
Test method
The data for gender is categorical and for height the datais ratio. The sample is effectively split into 2 sub-sets for male and female.
Most books give the independent samples t-test as the mainmethod for testing this hypothesis e.g. Curwin, et al (2001), Swift (2001).
Bland (2000) states that in order to use this test thesamples must both be from a normal populations and additionally thedistributions must have the same variance. Bland also suggests modifications tothe test when the variances cannot be assumed to be the same. Programs likeSPSS will calculate both for equal and non-equal variances. SPSS also gives atest for equality of variances.
When the assumptions of normality and independence are metthen the t-test is the best test according to Bland because it has higher powerthan the equivalent non-parametric test which is the Mann-Whitney U-test.However, the Mann-Whitney test is more robust in that it does not assume thatthe data is normally distributed.
It is a matter of weighing up the pros and cons. Ifnormality can be assumed then the independent samples t-test is best. If not,then the U-test should be used. Tests suchj as a histogram or Q-Q(quantile-quantile) plot can be used to check normality to help the decisionBland (2000).
Because the test is one-sided we would be looking for themale mean to be higher and the critical value to come from 0.05 in the one tailof the distribution. For the t test this would be looked up with n1 + n2 - 2degrees of freedom where n1 and n2 are the numbers of males and femalesrespectively.
It is also useful to work out a 95% confidence interval forthe population mean. This gives an idea of the spread of the estimate. Largersample sizes will reduce the confidence interval.
It was mentioned above that the inferences made are onlyvalid for the population being sampled and only so if the sample isrepresentative, which means selecting the sample from the whole population suchthat each member has equal probability of selection.
For the results to be reliable as Coolican (1990) says thatif a research finding can be repeated it is reliable. So, if the sample isrepeated the same result would indicate reliability.
Hypothesis 2: Taller children are heavier.
The null hypothesis is that there is no relationship betweenhow tall children are and how much they weigh. The alternative hypothesis isthat taller children are heavier, which is a one-sided test. That is, thealternative is not simply that there is a relationship, which would betwo-sided.
Both heights and weights are ratio data. This enables thedata to be examined by tests where normality is an underlying assumption.
In order to visually check the relationship a scatter graphis pretty well essential. This would give an idea of the strength and nature ofthe relationship. The relationship may not be linear as is often assumed. If sothen the scatter should show indication of a curve.
The strength of the relationship can be tested by using thePearson correlation coefficient ( r ). This is closely related to a regressionanalysis which would be fitting a straight line equation to the data withheight being the independent (x) variable and weight being dependent (y).
The correlation coefficient can be tested using a 1 sidedt-test. This has n-2 degrees of freedom, 28 in this case. The value of r wouldneed to be positive to indicate that taller children are heavier.
Analysis of the regression residuals can give us a lot moreinformation than simply carrying out a correlation calculation. See Bland(2000). They can be plotted to see whether they are normally distributed usinga histogram or Q-Q plot. Also, non-linearity should be apparent if this is thecase.
If the data shows a non-linear relationship then it would benecessary to transform the data using logs or other mathematical functions. Thetransformed variables would then need to be analyses for normality andlinearity.
According to Bland there is an alternative to the Pearsoncorrelation coefficient which does not assume that the data is normallydistributed. This is the Spearman Rank Correlation Coefficient. This is basedon the distribution of the ranks of the data and not the data itself. Thismakes it more robust in terms of departures from assumptions, however it isless powerful. In other words there is more chance of making a Type II error.
Again, if the sample is repeated the same result wouldindicate reliability.
Summary
The stages that need to be gone through in order to testhypotheses such as those above is as follows.
  • Definethe population about which inferences are to be made. This acts as a basis fromwhich to obtain a sample. The results will only apply to this population.
  • Decidehow to obtain a representative sample from the population. Decide the sample sizerequired and whether the sample can be stratified to make it more accurate.There is a large amount of literature available to help with sampling.
  • Setup null and alternative hypotheses giving particular attention to whether thealternative should be one or two sided.
  • Decideon the significance level before calculating the test statistic. This isusually 0.05 but sometimes 0.01. To be objective the value must be set beforecarrying out the analysis.
  • Choosethe testing method. Particular emphasis should be placed on the assumptionsthat the rest requires, particularly the assumption of normality which isneeded for tests like t tests. The pros and cons of the various alternativesshould be weighed up. This often boils down to power versus robustness.
  • Checknormality and linearity where appropriate by drawing graphs like QQ andhistograms for normality and scatter graphs for exploring relationships.
  • Calculatethe test statistic and compare with the critical value. Also calculate theprobability of obtaining the sample value. Reject the null hypothesis if thesample value is outside the critical range or single value if it is a one-sidedtest.
  • Beaware of the possibility of a Type II error which is accepting the nullhypothesis when it is in fact false.
References
Bland, J.Martin (2000) An Introduction to MedicalStatistics 3rd Edition Oxford. Oxford Medical Publications.
Curwin, Jon and Slater, Roger (2001) QuantitativeMethods for Business Decisions
London, Thomson Learning
Coolican, Hugh (1990) Research Methods and Statistics inPsychology London, Hodder and Stoughton
Swift, Louise (2001) Quantitative Methods for Business,Management and Finance, Basingstoke, Palgrave
 
审议和讨论所需的做法到fullanalysis的数据集提供。
作为这项工作的一部分,也探讨如何将测试下面的假设和解释的理由为自己的决定。假设1:男的孩子都长得比女性儿童。零假设有没有男性儿童和女性儿童之间的高度差。假设2:身材较高的子女也较重。零假设:有没有关系如何高大的孩子们,他们的体重多少。
分析数据集
该数据集是30个孩子的性别,年龄,身高,体重,上下肢的长度,眼睛的颜色,喜欢的巧克力或不andIQ的列表。
有两种主要的事情要考虑,分析捜之前。这些类型的数据和作为试样的数据的质量。
类型的数据可以是名义,有序,间隔或ratio.Nominal的分类也知道。 coolican (1990)给出了这些allof运算的更多的细节,他的定义已被用来决定海图集的数据的类型。
这也有利于区分,可以测量到任意数量的小数位数有限的跳跃,如1,2等整数,如一个discretenumbers continuousnumbers
性别
这个变量只能分辨男性或female.There之间没有为了这一点,所以数据是名义。
年龄
这一变量可以取整数值。这可能是measuredto小数,但一般只记录为整数。例如,它是比databecause ,它会是有意义的说,一个20岁的personis两次作为一个10岁的老。
在这组数据中,年龄范围从120个月156个月。这与被检测人群的需要是一致的。
高度
这个变量的值可以到小数点后ifnecessary 。这又是比数据,例如,因为这将是meaningfulto说,一个人谁是180厘米高是有人120cmtall高1.5倍。在此示例中,它是计算至最接近的cm 。
重量
喜欢的高度,这个变量可以被测量的小数位数比数据。在此示例中,它测量到最近的公斤。
上肢和下肢的长度
同样这个变量,如身高和体重,并是ratiodata 。
眼睛颜色
这一变量可以取的值为areeye颜色数目有限的。顺序没有意义。因此,这个数据是名义(分类) 。
喜欢的巧克力或不
与眼睛的颜色一样,可以采取这个变量的值数量有限的样本成员的喜好。在区分merelybetween喜欢和不喜欢中,顺序是没有意义的。此数据istherefore标称(分类) 。
智商
IQ是一个尺度测量通过测试每个samplemember发现。因此,它是不是比规模,因为它不会有意义tosay的,例如,有人得分125 , 100的得分是25%以上的智能thansomeone 。
还有另外一个级别的数据集变量拟合Cooligan intowhich没有提到的数据。这是有序数据。这意味着thatthe数据有一个订单或等级,这是有道理的。一个例子是,如果10students试图测试完成最快, 2ndquickest等,但不实际的时间记录。
数据乃拟从人口aboutwhich的是一个示例,我们可以作出推论。例如,在假设检验,我们希望无论他们是toknow人口差异的指标。结果canonly推断的人口从它是它不会是validotherwise绘制。
布兰德( 2000年)被发现在抽样方法详情。 Toaccomplish所要求的目标,样品有代表人口thedefined 。这也将是更准确的,如果采样stratifiedby已知的性别和年龄等因素。这意味着,例如,样品中的男性theproportion作为在thepopulation的比例是相同的。
样本大小是另一个考虑因素。在这种情况下,这是足够的,对于被测试的假说如下检查30.Whether 。
假设1:男性儿童长得比femalechildren 。
斯威夫特(2001)给出了一个非常可读的帐户的hypothesistesting过程和结构的测试。
第一步是成立的假设:
零假设是,在heightbetween男性儿童和女性儿童没有任何区别。
如果选择为Coolican把它描述为“我们做不预测结果将在哪个方向去,那么这将有beena双尾检验,在这种情况下,另一种方法是,男性有更高的它istherefore的一个特定的方向,所以一个尾检验是必需的。
为了检验这一假设,我们需要建立一个测试statisticand然后要么匹配它预先设定的临界值或calculatethe的概率实现采样值的基础上的假设,假设是正确的thenull 。
最常用的显着性水平为0.05 。根据斯威夫特(2001)显着性水平必须决定数据之前isknown 。这是停止调整的显着性水平的研究人员获得theresult他们想要的,而不是客观地接受或拒绝。
如果检验统计量的概率是小于0.05 ,我们wouldreject的零假设,没有任何区别男性片面的基础上重之间的男性andfemales的青睐。
然而,它有可能,测试统计在therejection区在事实上的零假设是真实的。这被称为一个TYPEI错误。
另外,也可以检验统计量时,在theacceptance区替代的假说是真实的(换句话说thenull假设为假) 。这就是所谓的第二类错误。功率为1一种Ⅱ型错误的概率,并因此correctlyrejecting虚假的零假设的概率。鉴于I型错误设置在thedesired的水平,第二类错误取决于实际值的alternativehypothesis 。
Coolican ( 1990年)载有可能的结果在下面的表:
在接受区
在排斥区
零假设为TRUE
I型错误
零假设为FALSE
第二类错误
测试方法
性别的数据分类和高度的datais比。样品被有效地分成2个子集,男性和女性。
大多数书籍给人的独立样本t检验测试这一假说,例如mainmethod Curwin等人(2001) ,斯威夫特(2001年) 。
布兰德(2000)指出,为了使用这种测试thesamples ,都必须从一个正常的人口和,附加thedistributions必须具有相同的方差。布兰德还建议修改tothe测试,不能假定方差是相同的。程序likeSPSS的都将计算为平等和非平等的差异。 SPSS也给测试平等的差异。
当正常和独立的假设metthen t检验是最好的测试根据布兰德,因为它具有较高的powerthan是采用Mann-Whitney U- test.However相当于非参数检验, Mann-Whitney检验强大,因为它不承担thatthe数据是正态分布的。
这是一个权衡的利弊问题。 Ifnormality可以假设,然后独立样本t检验是最好的。如果没有,那么U-检验应该被使用。测试suchj直方图或QQ (位数 - 分位数)的情节,可用于正常性检查,以帮助的decisionBland (2000) 。
由于测试是片面的我们将寻找themale ,意味着更高的临界值0.05在一个tailof分布。对于t检验,这将抬头N1 + N2 - 2degrees公司的自由,其中n1和n2的男性和femalesrespectively的是数字。
它也是有用的工作, 95 %的置信区间满心欢喜人口平均。这给出了一个理想的传播估计。 Largersample规模将减少的置信区间。
上述所作的推论onlyvalid被采样的人口只有如此,如果样品isrepresentative ,这意味着选择样本的整个人口suchthat的每个成员都有相同的概率选择。
结果是可靠的为Coolican (1990)说,它是可靠的,可重复的研究发现thatif 。所以,如果样品isrepeated相同的结果将表明其可靠性。
假设2:身材较高的子女也较重。
零假设是,有没有关系betweenhow高大的孩子们和他们的体重。替代假设isthat的身材较高的子女重,这是一种片面的测试。即, thealternative是不能简单地认为是有关系的, betwo双面。
身高和体重比数据。这使海图测试常态一个潜在的假设进行审查。
为了直观检查的分散GRAPHIS相当不错重要的关系。这将使的强度和性质的OFTHE关系的一个想法。这种关系可能不是线性的,如通常假定。如果sothen分散应该显示曲线的迹象。
强度的关系可以测试使用thePearson相关系数(r) 。这是密切相关的一个regressionanalysis将拟合直线方程是独立的(x)的变量和权重取决于数据withheight (y)的。
使用1 sidedt测试的可测试的相关系数。这具有在这种情况下,自由, 28度的n-2 。 r的值wouldneed是正的,表明身高的儿童重。
回归残差的分析可以给我们带来很多更多的信息比单纯进行相关计算。布兰德(2000年) 。他们可以绘制,看他们是否是正态分布usinga直方图或QQ图。另外,非线性应该是显而易见的,如果这是thecase 。
如果数据显示一个非线性的关系,那么这将benecessary使用日志或其他数学函数来转换数据。 Thetransformed变量,然后将需要要分析常态andlinearity的。
据布兰德还存在另一种的Pearsoncorrelation系数不承担数据normallydistributed的。这是Spearman等级相关系数。这是基于数据分布的行列,而不是数据本身。假设偏离Thismakes更强大的,但它isless强大。换言之,有更多的机会使II型错误。
再次,如果重复样品的相同的结果wouldindicate的可靠性。
总结
需要经历,以如上述的testhypotheses的阶段,如下所述。
Definethe人口作出推论。这样的行为作为基础fromwhich中,得到样品。结果将只适用于这一人群。
从人口Decidehow获得具有代表性的样本。决定样品sizerequired和样品是否可以分层,使其更多accurate.There是大量的文献可以帮助采样。
设置空和替代假说给予特别注意是否thealternative应该是一个或两个片面。
Decideon显着性水平,然后再计算检验统计量。这个isusually的0.05 ,但有时是0.01 。要客观的价值必须设置了分析beforecarrying 。
Choosethe测试方法。应特别强调放置在assumptionsthat上,其余的需要,尤其是正态性假设t检验,测试,如isneeded 。的利弊权衡各种alternativesshould的。这往往归结为功率与鲁棒性。
在适当的情况下绘制图形像QQ andhistograms的探索关系的常态和散点图Checknormality和线性度。
Calculatethe检验统计量和临界值进行比较。也计算theprobability取得的样本值。拒绝虚无假设如果thesample值超出临界范围或单值,如果它是一个sidedtest 。
Beaware接受nullhypothesis时,它其实也是假的II型错误的可能性。
参考文献
布兰德, J.Martin (2000)介绍到MedicalStatistics第三版牛津。牛津大学医学刊物。
Curwin ,乔恩和Slater ,罗杰(2001) QuantitativeMethods的商业决策
伦敦,汤姆森学习出版集团
休Coolican (1990)研究方法和统计inPsychology伦敦,霍德和斯托顿
斯威夫特,路易斯(2001)的定量方法商业,管理及财务,贝辛斯托克,帕尔格雷夫Consider and discuss the required approach to fullanalysis of the data set provided.
As part of this explore also how you would test the hypothesis below and explain the reasons for your decisions. Hypothesis 1: Male children are taller than female children. Null hypothesis; There is no difference in height between male children and female children. Hypothesis 2: Taller children are heavier. Null hypothesis: There is no relationship between how tall children are and how much they weigh.
Analysis of data set
The data set is a list of 30 children's gender, age, height,weight, upper and lower limb lengths, eye colour, like of chocolate or not andIQ.
There are two main things to consider before analysing thedata. These are the types of data and the quality of the data as a sample.
Types of data could be nominal, ordinal, interval or ratio.Nominal is also know as categorical. Coolican (1990) gives more details of allof these and his definitions have been used to decide the types of data in thedata set.
It is also helpful to distinguish between continuousnumbers, which could be measured to any number of decimal places an discretenumbers such as integers which have finite jumps like 1,2 etc.
Gender
This variable can only distinguish between male or female.There is no order to this and so the data is nominal.
Age
This variable can take integer values. It could be measuredto decimal places, but is generally only recorded as integer. It is ratio databecause, for example, it would be meaningful to say that a 20 year old personis twice as old as a 10 year old.
In this data set, the ages range from 120 months to 156months. This needs to be consistent with the population being tested.
Height
This variable can take values to decimal places ifnecessary. Again it is ratio data because, for example, it would be meaningfulto say that a person who is 180 cm tall is 1.5 times as tall as someone 120cmtall. In this sample it is measured to the nearest cm.
Weight
Like height, this variable could take be measured to decimalplaces and is ratio data. In this sample it is measured to the nearest kg.
Upper and lower limb lengths
Again this variable is like height and weight and is ratiodata.
Eye colour
This variable can take a limited number of values which areeye colours. The order is not meaningful. This data is therefore nominal(categorical).
Like of chocolate or not
As with eye colour, this variable can take a limited numberof values which are the sample members preferences. In distinguishing merelybetween liking and disliking, the order is not meaningful. This data istherefore nominal (categorical).
IQ
IQ is a scale measurement found by testing each samplemember. As such it is not a ratio scale because it would not be meaningful tosay, for example, that someone with a score of 125 is 25% more intelligent thansomeone with a score of 100.
There is another level of data mentioned by Cooligan intowhich none of the data set variables fit. That is Ordinal Data. This means thatthe data have an order or rank which makes sense. An example would be if 10students tried a test and you recorded who finished quickest, 2ndquickest etc, but not the actual time.
The data is intended to be a sample from a population aboutwhich we can make inferences. For example in the hypothesis tests we want toknow whether they are indicative of population differences. The results canonly be inferred on the population from which it is drawn it would not be validotherwise.
Details of sampling methods were found in Bland (2000). Toaccomplish the required objectives, the sample has to be representative of thedefined population. It would also be more accurate if the sample is stratifiedby known factors like gender and age. This means that, for example, theproportion of males in the sample is the same as the proportion in thepopulation.
Sample size is another consideration. In this case it is 30.Whether this is adequate for the hypotheses being tested is examined below.
Hypothesis 1: Male children are taller than femalechildren.
Swift (2001) gives a very readable account of the hypothesistesting process and the structure of the test.
The first step is to set up the hypotheses:
The Null hypothesis is that there is no difference in heightbetween male children and female children.
If the alternative was as Coolican describes it as "wedo not predict in which direction the results will go then it would have beena two-tailed test. In this case the alternative is that males are taller it istherefore a specific direction and so a one-tailed test is required.
To test the hypothesis we need to set up a test statisticand then either match it against a pre-determined critical value or calculatethe probability of achieving the sample value based on the assumption that thenull hypothesis is true.
The most commonly used significance level is 0.05. Accordingto Swift (2001) the significance level must be decided before the data isknown. This is to stop researchers adjusting the significance level to get theresult that they want rather than accepting or rejecting objectively.
If the test statistic probability is less than 0.05 we wouldreject the null hypothesis that there is no difference between males andfemales in favour of males being heavier on the one sided basis.
However it is possible for the test statistic to be in therejection zone when in fact the null hypothesis is true. This is called a TypeI error.
It is also possible for the test statistic to be in theacceptance zone when the alternative hypothesis is true (in other words thenull hypothesis is false). This is called a Type II error. Power is 1 -probability of a Type II error and is therefore the probability of correctlyrejecting a false null hypothesis. Whereas the Type I error is set at thedesired level, the Type II error depends on the actual value of the alternativehypothesis.
Coolican (1990) sets out the possible outcomes in thefollowing table:
  In acceptance zone In rejection zone
NULL Hypothesis TRUE OK Type I error
NULL Hypothesis FALSE Type II error OK
Test method
The data for gender is categorical and for height the datais ratio. The sample is effectively split into 2 sub-sets for male and female.
Most books give the independent samples t-test as the mainmethod for testing this hypothesis e.g. Curwin, et al (2001), Swift (2001).
Bland (2000) states that in order to use this test thesamples must both be from a normal populations and additionally thedistributions must have the same variance. Bland also suggests modifications tothe test when the variances cannot be assumed to be the same. Programs likeSPSS will calculate both for equal and non-equal variances. SPSS also gives atest for equality of variances.
When the assumptions of normality and independence are metthen the t-test is the best test according to Bland because it has higher powerthan the equivalent non-parametric test which is the Mann-Whitney U-test.However, the Mann-Whitney test is more robust in that it does not assume thatthe data is normally distributed.
It is a matter of weighing up the pros and cons. Ifnormality can be assumed then the independent samples t-test is best. If not,then the U-test should be used. Tests suchj as a histogram or Q-Q(quantile-quantile) plot can be used to check normality to help the decisionBland (2000).
Because the test is one-sided we would be looking for themale mean to be higher and the critical value to come from 0.05 in the one tailof the distribution. For the t test this would be looked up with n1 + n2 - 2degrees of freedom where n1 and n2 are the numbers of males and femalesrespectively.
It is also useful to work out a 95% confidence interval forthe population mean. This gives an idea of the spread of the estimate. Largersample sizes will reduce the confidence interval.
It was mentioned above that the inferences made are onlyvalid for the population being sampled and only so if the sample isrepresentative, which means selecting the sample from the whole population suchthat each member has equal probability of selection.
For the results to be reliable as Coolican (1990) says thatif a research finding can be repeated it is reliable. So, if the sample isrepeated the same result would indicate reliability.
Hypothesis 2: Taller children are heavier.
The null hypothesis is that there is no relationship betweenhow tall children are and how much they weigh. The alternative hypothesis isthat taller children are heavier, which is a one-sided test. That is, thealternative is not simply that there is a relationship, which would betwo-sided.
Both heights and weights are ratio data. This enables thedata to be examined by tests where normality is an underlying assumption.
In order to visually check the relationship a scatter graphis pretty well essential. This would give an idea of the strength and nature ofthe relationship. The relationship may not be linear as is often assumed. If sothen the scatter should show indication of a curve.
The strength of the relationship can be tested by using thePearson correlation coefficient ( r ). This is closely related to a regressionanalysis which would be fitting a straight line equation to the data withheight being the independent (x) variable and weight being dependent (y).
The correlation coefficient can be tested using a 1 sidedt-test. This has n-2 degrees of freedom, 28 in this case. The value of r wouldneed to be positive to indicate that taller children are heavier.
Analysis of the regression residuals can give us a lot moreinformation than simply carrying out a correlation calculation. See Bland(2000). They can be plotted to see whether they are normally distributed usinga histogram or Q-Q plot. Also, non-linearity should be apparent if this is thecase.
If the data shows a non-linear relationship then it would benecessary to transform the data using logs or other mathematical functions. Thetransformed variables would then need to be analyses for normality andlinearity.
According to Bland there is an alternative to the Pearsoncorrelation coefficient which does not assume that the data is normallydistributed. This is the Spearman Rank Correlation Coefficient. This is basedon the distribution of the ranks of the data and not the data itself. Thismakes it more robust in terms of departures from assumptions, however it isless powerful. In other words there is more chance of making a Type II error.
Again, if the sample is repeated the same result wouldindicate reliability.
Summary
The stages that need to be gone through in order to testhypotheses such as those above is as follows.
  • Definethe population about which inferences are to be made. This acts as a basis fromwhich to obtain a sample. The results will only apply to this population.
  • Decidehow to obtain a representative sample from the population. Decide the sample sizerequired and whether the sample can be stratified to make it more accurate.There is a large amount of literature available to help with sampling.
  • Setup null and alternative hypotheses giving particular attention to whether thealternative should be one or two sided.
  • Decideon the significance level before calculating the test statistic. This isusually 0.05 but sometimes 0.01. To be objective the value must be set beforecarrying out the analysis.
  • Choosethe testing method. Particular emphasis should be placed on the assumptionsthat the rest requires, particularly the assumption of normality which isneeded for tests like t tests. The pros and cons of the various alternativesshould be weighed up. This often boils down to power versus robustness.
  • Checknormality and linearity where appropriate by drawing graphs like QQ andhistograms for normality and scatter graphs for exploring relationships.
  • Calculatethe test statistic and compare with the critical value. Also calculate theprobability of obtaining the sample value. Reject the null hypothesis if thesample value is outside the critical range or single value if it is a one-sidedtest.
  • Beaware of the possibility of a Type II error which is accepting the nullhypothesis when it is in fact false.
References
Bland, J.Martin (2000) An Introduction to MedicalStatistics 3rd Edition Oxford. Oxford Medical Publications.
Curwin, Jon and Slater, Roger (2001) QuantitativeMethods for Business Decisions
London, Thomson Learning
Coolican, Hugh (1990) Research Methods and Statistics inPsychology London, Hodder and Stoughton
Swift, Louise (2001) Quantitative Methods for Business,Management and Finance, Basingstoke, Palgrave
 
审议和讨论所需的做法到fullanalysis的数据集提供。
作为这项工作的一部分,也探讨如何将测试下面的假设和解释的理由为自己的决定。假设1:男的孩子都长得比女性儿童。零假设有没有男性儿童和女性儿童之间的高度差。假设2:身材较高的子女也较重。零假设:有没有关系如何高大的孩子们,他们的体重多少。
分析数据集
该数据集是30个孩子的性别,年龄,身高,体重,上下肢的长度,眼睛的颜色,喜欢的巧克力或不andIQ的列表。
有两种主要的事情要考虑,分析捜之前。这些类型的数据和作为试样的数据的质量。
类型的数据可以是名义,有序,间隔或ratio.Nominal的分类也知道。 coolican (1990)给出了这些allof运算的更多的细节,他的定义已被用来决定海图集的数据的类型。
这也有利于区分,可以测量到任意数量的小数位数有限的跳跃,如1,2等整数,如一个discretenumbers continuousnumbers
性别
这个变量只能分辨男性或female.There之间没有为了这一点,所以数据是名义。
年龄
这一变量可以取整数值。这可能是measuredto小数,但一般只记录为整数。例如,它是比databecause ,它会是有意义的说,一个20岁的personis两次作为一个10岁的老。
在这组数据中,年龄范围从120个月156个月。这与被检测人群的需要是一致的。
高度
这个变量的值可以到小数点后ifnecessary 。这又是比数据,例如,因为这将是meaningfulto说,一个人谁是180厘米高是有人120cmtall高1.5倍。在此示例中,它是计算至最接近的cm 。
重量
喜欢的高度,这个变量可以被测量的小数位数比数据。在此示例中,它测量到最近的公斤。
上肢和下肢的长度
同样这个变量,如身高和体重,并是ratiodata 。
眼睛颜色
这一变量可以取的值为areeye颜色数目有限的。顺序没有意义。因此,这个数据是名义(分类) 。
喜欢的巧克力或不
与眼睛的颜色一样,可以采取这个变量的值数量有限的样本成员的喜好。在区分merelybetween喜欢和不喜欢中,顺序是没有意义的。此数据istherefore标称(分类) 。
智商
IQ是一个尺度测量通过测试每个samplemember发现。因此,它是不是比规模,因为它不会有意义tosay的,例如,有人得分125 , 100的得分是25%以上的智能thansomeone 。
还有另外一个级别的数据集变量拟合Cooligan intowhich没有提到的数据。这是有序数据。这意味着thatthe数据有一个订单或等级,这是有道理的。一个例子是,如果10students试图测试完成最快, 2ndquickest等,但不实际的时间记录。
数据乃拟从人口aboutwhich的是一个示例,我们可以作出推论。例如,在假设检验,我们希望无论他们是toknow人口差异的指标。结果canonly推断的人口从它是它不会是validotherwise绘制。
布兰德( 2000年)被发现在抽样方法详情。 Toaccomplish所要求的目标,样品有代表人口thedefined 。这也将是更准确的,如果采样stratifiedby已知的性别和年龄等因素。这意味着,例如,样品中的男性theproportion作为在thepopulation的比例是相同的。
样本大小是另一个考虑因素。在这种情况下,这是足够的,对于被测试的假说如下检查30.Whether 。
假设1:男性儿童长得比femalechildren 。
斯威夫特(2001)给出了一个非常可读的帐户的hypothesistesting过程和结构的测试。
第一步是成立的假设:
零假设是,在heightbetween男性儿童和女性儿童没有任何区别。
如果选择为Coolican把它描述为“我们做不预测结果将在哪个方向去,那么这将有beena双尾检验,在这种情况下,另一种方法是,男性有更高的它istherefore的一个特定的方向,所以一个尾检验是必需的。
为了检验这一假设,我们需要建立一个测试statisticand然后要么匹配它预先设定的临界值或calculatethe的概率实现采样值的基础上的假设,假设是正确的thenull 。
最常用的显着性水平为0.05 。根据斯威夫特(2001)显着性水平必须决定数据之前isknown 。这是停止调整的显着性水平的研究人员获得theresult他们想要的,而不是客观地接受或拒绝。
如果检验统计量的概率是小于0.05 ,我们wouldreject的零假设,没有任何区别男性片面的基础上重之间的男性andfemales的青睐。
然而,它有可能,测试统计在therejection区在事实上的零假设是真实的。这被称为一个TYPEI错误。
另外,也可以检验统计量时,在theacceptance区替代的假说是真实的(换句话说thenull假设为假) 。这就是所谓的第二类错误。功率为1一种Ⅱ型错误的概率,并因此correctlyrejecting虚假的零假设的概率。鉴于I型错误设置在thedesired的水平,第二类错误取决于实际值的alternativehypothesis 。
Coolican ( 1990年)载有可能的结果在下面的表:
在接受区
在排斥区
零假设为TRUE
I型错误
零假设为FALSE
第二类错误
测试方法
性别的数据分类和高度的datais比。样品被有效地分成2个子集,男性和女性。
大多数书籍给人的独立样本t检验测试这一假说,例如mainmethod Curwin等人(2001) ,斯威夫特(2001年) 。
布兰德(2000)指出,为了使用这种测试thesamples ,都必须从一个正常的人口和,附加thedistributions必须具有相同的方差。布兰德还建议修改tothe测试,不能假定方差是相同的。程序likeSPSS的都将计算为平等和非平等的差异。 SPSS也给测试平等的差异。
当正常和独立的假设metthen t检验是最好的测试根据布兰德,因为它具有较高的powerthan是采用Mann-Whitney U- test.However相当于非参数检验, Mann-Whitney检验强大,因为它不承担thatthe数据是正态分布的。
这是一个权衡的利弊问题。 Ifnormality可以假设,然后独立样本t检验是最好的。如果没有,那么U-检验应该被使用。测试suchj直方图或QQ (位数 - 分位数)的情节,可用于正常性检查,以帮助的decisionBland (2000) 。
由于测试是片面的我们将寻找themale ,意味着更高的临界值0.05在一个tailof分布。对于t检验,这将抬头N1 + N2 - 2degrees公司的自由,其中n1和n2的男性和femalesrespectively的是数字。
它也是有用的工作, 95 %的置信区间满心欢喜人口平均。这给出了一个理想的传播估计。 Largersample规模将减少的置信区间。
上述所作的推论onlyvalid被采样的人口只有如此,如果样品isrepresentative ,这意味着选择样本的整个人口suchthat的每个成员都有相同的概率选择。
结果是可靠的为Coolican (1990)说,它是可靠的,可重复的研究发现thatif 。所以,如果样品isrepeated相同的结果将表明其可靠性。
假设2:身材较高的子女也较重。
零假设是,有没有关系betweenhow高大的孩子们和他们的体重。替代假设isthat的身材较高的子女重,这是一种片面的测试。即, thealternative是不能简单地认为是有关系的, betwo双面。
身高和体重比数据。这使海图测试常态一个潜在的假设进行审查。
为了直观检查的分散GRAPHIS相当不错重要的关系。这将使的强度和性质的OFTHE关系的一个想法。这种关系可能不是线性的,如通常假定。如果sothen分散应该显示曲线的迹象。
强度的关系可以测试使用thePearson相关系数(r) 。这是密切相关的一个regressionanalysis将拟合直线方程是独立的(x)的变量和权重取决于数据withheight (y)的。
使用1 sidedt测试的可测试的相关系数。这具有在这种情况下,自由, 28度的n-2 。 r的值wouldneed是正的,表明身高的儿童重。
回归残差的分析可以给我们带来很多更多的信息比单纯进行相关计算。布兰德(2000年) 。他们可以绘制,看他们是否是正态分布usinga直方图或QQ图。另外,非线性应该是显而易见的,如果这是thecase 。
如果数据显示一个非线性的关系,那么这将benecessary使用日志或其他数学函数来转换数据。 Thetransformed变量,然后将需要要分析常态andlinearity的。
据布兰德还存在另一种的Pearsoncorrelation系数不承担数据normallydistributed的。这是Spearman等级相关系数。这是基于数据分布的行列,而不是数据本身。假设偏离Thismakes更强大的,但它isless强大。换言之,有更多的机会使II型错误。
再次,如果重复样品的相同的结果wouldindicate的可靠性。
总结
需要经历,以如上述的testhypotheses的阶段,如下所述。
Definethe人口作出推论。这样的行为作为基础fromwhich中,得到样品。结果将只适用于这一人群。
从人口Decidehow获得具有代表性的样本。决定样品sizerequired和样品是否可以分层,使其更多accurate.There是大量的文献可以帮助采样。
设置空和替代假说给予特别注意是否thealternative应该是一个或两个片面。
Decideon显着性水平,然后再计算检验统计量。这个isusually的0.05 ,但有时是0.01 。要客观的价值必须设置了分析beforecarrying 。
Choosethe测试方法。应特别强调放置在assumptionsthat上,其余的需要,尤其是正态性假设t检验,测试,如isneeded 。的利弊权衡各种alternativesshould的。这往往归结为功率与鲁棒性。
在适当的情况下绘制图形像QQ andhistograms的探索关系的常态和散点图Checknormality和线性度。
Calculatethe检验统计量和临界值进行比较。也计算theprobability取得的样本值。拒绝虚无假设如果thesample值超出临界范围或单值,如果它是一个sidedtest 。
Beaware接受nullhypothesis时,它其实也是假的II型错误的可能性。
参考文献
布兰德, J.Martin (2000)介绍到MedicalStatistics第三版牛津。牛津大学医学刊物。
Curwin ,乔恩和Slater ,罗杰(2001) QuantitativeMethods的商业决策
伦敦,汤姆森学习出版集团
休Coolican (1990)研究方法和统计inPsychology伦敦,霍德和斯托顿
斯威夫特,路易斯(2001)的定量方法商业,管理及财务,贝辛斯托克,帕尔格雷夫Consider and discuss the required approach to fullanalysis of the data set provided.
As part of this explore also how you would test the hypothesis below and explain the reasons for your decisions. Hypothesis 1: Male children are taller than female children. Null hypothesis; There is no difference in height between male children and female children. Hypothesis 2: Taller children are heavier. Null hypothesis: There is no relationship between how tall children are and how much they weigh.
Analysis of data set
The data set is a list of 30 children's gender, age, height,weight, upper and lower limb lengths, eye colour, like of chocolate or not andIQ.
There are two main things to consider before analysing thedata. These are the types of data and the quality of the data as a sample.
Types of data could be nominal, ordinal, interval or ratio.Nominal is also know as categorical. Coolican (1990) gives more details of allof these and his definitions have been used to decide the types of data in thedata set.
It is also helpful to distinguish between continuousnumbers, which could be measured to any number of decimal places an discretenumbers such as integers which have finite jumps like 1,2 etc.
Gender
This variable can only distinguish between male or female.There is no order to this and so the data is nominal.
Age
This variable can take integer values. It could be measuredto decimal places, but is generally only recorded as integer. It is ratio databecause, for example, it would be meaningful to say that a 20 year old personis twice as old as a 10 year old.
In this data set, the ages range from 120 months to 156months. This needs to be consistent with the population being tested.
Height
This variable can take values to decimal places ifnecessary. Again it is ratio data because, for example, it would be meaningfulto say that a person who is 180 cm tall is 1.5 times as tall as someone 120cmtall. In this sample it is measured to the nearest cm.
Weight
Like height, this variable could take be measured to decimalplaces and is ratio data. In this sample it is measured to the nearest kg.
Upper and lower limb lengths
Again this variable is like height and weight and is ratiodata.
Eye colour
This variable can take a limited number of values which areeye colours. The order is not meaningful. This data is therefore nominal(categorical).
Like of chocolate or not
As with eye colour, this variable can take a limited numberof values which are the sample members preferences. In distinguishing merelybetween liking and disliking, the order is not meaningful. This data istherefore nominal (categorical).
IQ
IQ is a scale measurement found by testing each samplemember. As such it is not a ratio scale because it would not be meaningful tosay, for example, that someone with a score of 125 is 25% more intelligent thansomeone with a score of 100.
There is another level of data mentioned by Cooligan intowhich none of the data set variables fit. That is Ordinal Data. This means thatthe data have an order or rank which makes sense. An example would be if 10students tried a test and you recorded who finished quickest, 2ndquickest etc, but not the actual time.
The data is intended to be a sample from a population aboutwhich we can make inferences. For example in the hypothesis tests we want toknow whether they are indicative of population differences. The results canonly be inferred on the population from which it is drawn it would not be validotherwise.
Details of sampling methods were found in Bland (2000). Toaccomplish the required objectives, the sample has to be representative of thedefined population. It would also be more accurate if the sample is stratifiedby known factors like gender and age. This means that, for example, theproportion of males in the sample is the same as the proportion in thepopulation.
Sample size is another consideration. In this case it is 30.Whether this is adequate for the hypotheses being tested is examined below.
Hypothesis 1: Male children are taller than femalechildren.
Swift (2001) gives a very readable account of the hypothesistesting process and the structure of the test.
The first step is to set up the hypotheses:
The Null hypothesis is that there is no difference in heightbetween male children and female children.
If the alternative was as Coolican describes it as "wedo not predict in which direction the results will go then it would have beena two-tailed test. In this case the alternative is that males are taller it istherefore a specific direction and so a one-tailed test is required.
To test the hypothesis we need to set up a test statisticand then either match it against a pre-determined critical value or calculatethe probability of achieving the sample value based on the assumption that thenull hypothesis is true.
The most commonly used significance level is 0.05. Accordingto Swift (2001) the significance level must be decided before the data isknown. This is to stop researchers adjusting the significance level to get theresult that they want rather than accepting or rejecting objectively.
If the test statistic probability is less than 0.05 we wouldreject the null hypothesis that there is no difference between males andfemales in favour of males being heavier on the one sided basis.
However it is possible for the test statistic to be in therejection zone when in fact the null hypothesis is true. This is called a TypeI error.
It is also possible for the test statistic to be in theacceptance zone when the alternative hypothesis is true (in other words thenull hypothesis is false). This is called a Type II error. Power is 1 -probability of a Type II error and is therefore the probability of correctlyrejecting a false null hypothesis. Whereas the Type I error is set at thedesired level, the Type II error depends on the actual value of the alternativehypothesis.
Coolican (1990) sets out the possible outcomes in thefollowing table:
  In acceptance zone In rejection zone
NULL Hypothesis TRUE OK Type I error
NULL Hypothesis FALSE Type II error OK
Test method
The data for gender is categorical and for height the datais ratio. The sample is effectively split into 2 sub-sets for male and female.
Most books give the independent samples t-test as the mainmethod for testing this hypothesis e.g. Curwin, et al (2001), Swift (2001).
Bland (2000) states that in order to use this test thesamples must both be from a normal populations and additionally thedistributions must have the same variance. Bland also suggests modifications tothe test when the variances cannot be assumed to be the same. Programs likeSPSS will calculate both for equal and non-equal variances. SPSS also gives atest for equality of variances.
When the assumptions of normality and independence are metthen the t-test is the best test according to Bland because it has higher powerthan the equivalent non-parametric test which is the Mann-Whitney U-test.However, the Mann-Whitney test is more robust in that it does not assume thatthe data is normally distributed.
It is a matter of weighing up the pros and cons. Ifnormality can be assumed then the independent samples t-test is best. If not,then the U-test should be used. Tests suchj as a histogram or Q-Q(quantile-quantile) plot can be used to check normality to help the decisionBland (2000).
Because the test is one-sided we would be looking for themale mean to be higher and the critical value to come from 0.05 in the one tailof the distribution. For the t test this would be looked up with n1 + n2 - 2degrees of freedom where n1 and n2 are the numbers of males and femalesrespectively.
It is also useful to work out a 95% confidence interval forthe population mean. This gives an idea of the spread of the estimate. Largersample sizes will reduce the confidence interval.
It was mentioned above that the inferences made are onlyvalid for the population being sampled and only so if the sample isrepresentative, which means selecting the sample from the whole population suchthat each member has equal probability of selection.
For the results to be reliable as Coolican (1990) says thatif a research finding can be repeated it is reliable. So, if the sample isrepeated the same result would indicate reliability.
Hypothesis 2: Taller children are heavier.
The null hypothesis is that there is no relationship betweenhow tall children are and how much they weigh. The alternative hypothesis isthat taller children are heavier, which is a one-sided test. That is, thealternative is not simply that there is a relationship, which would betwo-sided.
Both heights and weights are ratio data. This enables thedata to be examined by tests where normality is an underlying assumption.
In order to visually check the relationship a scatter graphis pretty well essential. This would give an idea of the strength and nature ofthe relationship. The relationship may not be linear as is often assumed. If sothen the scatter should show indication of a curve.
The strength of the relationship can be tested by using thePearson correlation coefficient ( r ). This is closely related to a regressionanalysis which would be fitting a straight line equation to the data withheight being the independent (x) variable and weight being dependent (y).
The correlation coefficient can be tested using a 1 sidedt-test. This has n-2 degrees of freedom, 28 in this case. The value of r wouldneed to be positive to indicate that taller children are heavier.
Analysis of the regression residuals can give us a lot moreinformation than simply carrying out a correlation calculation. See Bland(2000). They can be plotted to see whether they are normally distributed usinga histogram or Q-Q plot. Also, non-linearity should be apparent if this is thecase.
If the data shows a non-linear relationship then it would benecessary to transform the data using logs or other mathematical functions. Thetransformed variables would then need to be analyses for normality andlinearity.
According to Bland there is an alternative to the Pearsoncorrelation coefficient which does not assume that the data is normallydistributed. This is the Spearman Rank Correlation Coefficient. This is basedon the distribution of the ranks of the data and not the data itself. Thismakes it more robust in terms of departures from assumptions, however it isless powerful. In other words there is more chance of making a Type II error.
Again, if the sample is repeated the same result wouldindicate reliability.
Summary
The stages that need to be gone through in order to testhypotheses such as those above is as follows.
  • Definethe population about which inferences are to be made. This acts as a basis fromwhich to obtain a sample. The results will only apply to this population.
  • Decidehow to obtain a representative sample from the population. Decide the sample sizerequired and whether the sample can be stratified to make it more accurate.There is a large amount of literature available to help with sampling.
  • Setup null and alternative hypotheses giving particular attention to whether thealternative should be one or two sided.
  • Decideon the significance level before calculating the test statistic. This isusually 0.05 but sometimes 0.01. To be objective the value must be set beforecarrying out the analysis.
  • Choosethe testing method. Particular emphasis should be placed on the assumptionsthat the rest requires, particularly the assumption of normality which isneeded for tests like t tests. The pros and cons of the various alternativesshould be weighed up. This often boils down to power versus robustness.
  • Checknormality and linearity where appropriate by drawing graphs like QQ andhistograms for normality and scatter graphs for exploring relationships.
  • Calculatethe test statistic and compare with the critical value. Also calculate theprobability of obtaining the sample value. Reject the null hypothesis if thesample value is outside the critical range or single value if it is a one-sidedtest.
  • Beaware of the possibility of a Type II error which is accepting the nullhypothesis when it is in fact false.
References
Bland, J.Martin (2000) An Introduction to MedicalStatistics 3rd Edition Oxford. Oxford Medical Publications.
Curwin, Jon and Slater, Roger (2001) QuantitativeMethods for Business Decisions
London, Thomson Learning
Coolican, Hugh (1990) Research Methods and Statistics inPsychology London, Hodder and Stoughton
Swift, Louise (2001) Quantitative Methods for Business,Management and Finance, Basingstoke, Palgrave
 
审议和讨论所需的做法到fullanalysis的数据集提供。
作为这项工作的一部分,也探讨如何将测试下面的假设和解释的理由为自己的决定。假设1:男的孩子都长得比女性儿童。零假设有没有男性儿童和女性儿童之间的高度差。假设2:身材较高的子女也较重。零假设:有没有关系如何高大的孩子们,他们的体重多少。
分析数据集
该数据集是30个孩子的性别,年龄,身高,体重,上下肢的长度,眼睛的颜色,喜欢的巧克力或不andIQ的列表。
有两种主要的事情要考虑,分析捜之前。这些类型的数据和作为试样的数据的质量。
类型的数据可以是名义,有序,间隔或ratio.Nominal的分类也知道。 coolican (1990)给出了这些allof运算的更多的细节,他的定义已被用来决定海图集的数据的类型。
这也有利于区分,可以测量到任意数量的小数位数有限的跳跃,如1,2等整数,如一个discretenumbers continuousnumbers
性别
这个变量只能分辨男性或female.There之间没有为了这一点,所以数据是名义。
年龄
这一变量可以取整数值。这可能是measuredto小数,但一般只记录为整数。例如,它是比databecause ,它会是有意义的说,一个20岁的personis两次作为一个10岁的老。
在这组数据中,年龄范围从120个月156个月。这与被检测人群的需要是一致的。
高度
这个变量的值可以到小数点后ifnecessary 。这又是比数据,例如,因为这将是meaningfulto说,一个人谁是180厘米高是有人120cmtall高1.5倍。在此示例中,它是计算至最接近的cm 。
重量
喜欢的高度,这个变量可以被测量的小数位数比数据。在此示例中,它测量到最近的公斤。
上肢和下肢的长度
同样这个变量,如身高和体重,并是ratiodata 。
眼睛颜色
这一变量可以取的值为areeye颜色数目有限的。顺序没有意义。因此,这个数据是名义(分类) 。
喜欢的巧克力或不
与眼睛的颜色一样,可以采取这个变量的值数量有限的样本成员的喜好。在区分merelybetween喜欢和不喜欢中,顺序是没有意义的。此数据istherefore标称(分类) 。
智商
IQ是一个尺度测量通过测试每个samplemember发现。因此,它是不是比规模,因为它不会有意义tosay的,例如,有人得分125 , 100的得分是25%以上的智能thansomeone 。
还有另外一个级别的数据集变量拟合Cooligan intowhich没有提到的数据。这是有序数据。这意味着thatthe数据有一个订单或等级,这是有道理的。一个例子是,如果10students试图测试完成最快, 2ndquickest等,但不实际的时间记录。
数据乃拟从人口aboutwhich的是一个示例,我们可以作出推论。例如,在假设检验,我们希望无论他们是toknow人口差异的指标。结果canonly推断的人口从它是它不会是validotherwise绘制。
布兰德( 2000年)被发现在抽样方法详情。 Toaccomplish所要求的目标,样品有代表人口thedefined 。这也将是更准确的,如果采样stratifiedby已知的性别和年龄等因素。这意味着,例如,样品中的男性theproportion作为在thepopulation的比例是相同的。
样本大小是另一个考虑因素。在这种情况下,这是足够的,对于被测试的假说如下检查30.Whether 。
假设1:男性儿童长得比femalechildren 。
斯威夫特(2001)给出了一个非常可读的帐户的hypothesistesting过程和结构的测试。
第一步是成立的假设:
零假设是,在heightbetween男性儿童和女性儿童没有任何区别。
如果选择为Coolican把它描述为“我们做不预测结果将在哪个方向去,那么这将有beena双尾检验,在这种情况下,另一种方法是,男性有更高的它istherefore的一个特定的方向,所以一个尾检验是必需的。
为了检验这一假设,我们需要建立一个测试statisticand然后要么匹配它预先设定的临界值或calculatethe的概率实现采样值的基础上的假设,假设是正确的thenull 。
最常用的显着性水平为0.05 。根据斯威夫特(2001)显着性水平必须决定数据之前isknown 。这是停止调整的显着性水平的研究人员获得theresult他们想要的,而不是客观地接受或拒绝。
如果检验统计量的概率是小于0.05 ,我们wouldreject的零假设,没有任何区别男性片面的基础上重之间的男性andfemales的青睐。
然而,它有可能,测试统计在therejection区在事实上的零假设是真实的。这被称为一个TYPEI错误。
另外,也可以检验统计量时,在theacceptance区替代的假说是真实的(换句话说thenull假设为假) 。这就是所谓的第二类错误。功率为1一种Ⅱ型错误的概率,并因此correctlyrejecting虚假的零假设的概率。鉴于I型错误设置在thedesired的水平,第二类错误取决于实际值的alternativehypothesis 。
Coolican ( 1990年)载有可能的结果在下面的表:
在接受区
在排斥区
零假设为TRUE
I型错误
零假设为FALSE
第二类错误
测试方法
性别的数据分类和高度的datais比。样品被有效地分成2个子集,男性和女性。
大多数书籍给人的独立样本t检验测试这一假说,例如mainmethod Curwin等人(2001) ,斯威夫特(2001年) 。
布兰德(2000)指出,为了使用这种测试thesamples ,都必须从一个正常的人口和,附加thedistributions必须具有相同的方差。布兰德还建议修改tothe测试,不能假定方差是相同的。程序likeSPSS的都将计算为平等和非平等的差异。 SPSS也给测试平等的差异。
当正常和独立的假设metthen t检验是最好的测试根据布兰德,因为它具有较高的powerthan是采用Mann-Whitney U- test.However相当于非参数检验, Mann-Whitney检验强大,因为它不承担thatthe数据是正态分布的。
这是一个权衡的利弊问题。 Ifnormality可以假设,然后独立样本t检验是最好的。如果没有,那么U-检验应该被使用。测试suchj直方图或QQ (位数 - 分位数)的情节,可用于正常性检查,以帮助的decisionBland (2000) 。
由于测试是片面的我们将寻找themale ,意味着更高的临界值0.05在一个tailof分布。对于t检验,这将抬头N1 + N2 - 2degrees公司的自由,其中n1和n2的男性和femalesrespectively的是数字。
它也是有用的工作, 95 %的置信区间满心欢喜人口平均。这给出了一个理想的传播估计。 Largersample规模将减少的置信区间。
上述所作的推论onlyvalid被采样的人口只有如此,如果样品isrepresentative ,这意味着选择样本的整个人口suchthat的每个成员都有相同的概率选择。
结果是可靠的为Coolican (1990)说,它是可靠的,可重复的研究发现thatif 。所以,如果样品isrepeated相同的结果将表明其可靠性。
假设2:身材较高的子女也较重。
零假设是,有没有关系betweenhow高大的孩子们和他们的体重。替代假设isthat的身材较高的子女重,这是一种片面的测试。即, thealternative是不能简单地认为是有关系的, betwo双面。
身高和体重比数据。这使海图测试常态一个潜在的假设进行审查。
为了直观检查的分散GRAPHIS相当不错重要的关系。这将使的强度和性质的OFTHE关系的一个想法。这种关系可能不是线性的,如通常假定。如果sothen分散应该显示曲线的迹象。
强度的关系可以测试使用thePearson相关系数(r) 。这是密切相关的一个regressionanalysis将拟合直线方程是独立的(x)的变量和权重取决于数据withheight (y)的。
使用1 sidedt测试的可测试的相关系数。这具有在这种情况下,自由, 28度的n-2 。 r的值wouldneed是正的,表明身高的儿童重。
回归残差的分析可以给我们带来很多更多的信息比单纯进行相关计算。布兰德(2000年) 。他们可以绘制,看他们是否是正态分布usinga直方图或QQ图。另外,非线性应该是显而易见的,如果这是thecase 。
如果数据显示一个非线性的关系,那么这将benecessary使用日志或其他数学函数来转换数据。 Thetransformed变量,然后将需要要分析常态andlinearity的。
据布兰德还存在另一种的Pearsoncorrelation系数不承担数据normallydistributed的。这是Spearman等级相关系数。这是基于数据分布的行列,而不是数据本身。假设偏离Thismakes更强大的,但它isless强大。换言之,有更多的机会使II型错误。
再次,如果重复样品的相同的结果wouldindicate的可靠性。
总结
需要经历,以如上述的testhypotheses的阶段,如下所述。
Definethe人口作出推论。这样的行为作为基础fromwhich中,得到样品。结果将只适用于这一人群。
从人口Decidehow获得具有代表性的样本。决定样品sizerequired和样品是否可以分层,使其更多accurate.There是大量的文献可以帮助采样。
设置空和替代假说给予特别注意是否thealternative应该是一个或两个片面。
Decideon显着性水平,然后再计算检验统计量。这个isusually的0.05 ,但有时是0.01 。要客观的价值必须设置了分析beforecarrying 。
Choosethe测试方法。应特别强调放置在assumptionsthat上,其余的需要,尤其是正态性假设t检验,测试,如isneeded 。的利弊权衡各种alternativesshould的。这往往归结为功率与鲁棒性。
在适当的情况下绘制图形像QQ andhistograms的探索关系的常态和散点图Checknormality和线性度。
Calculatethe检验统计量和临界值进行比较。也计算theprobability取得的样本值。拒绝虚无假设如果thesample值超出临界范围或单值,如果它是一个sidedtest 。
Beaware接受nullhypothesis时,它其实也是假的II型错误的可能性。
参考文献
布兰德, J.Martin (2000)介绍到MedicalStatistics第三版牛津。牛津大学医学刊物。
Curwin ,乔恩和Slater ,罗杰(2001) QuantitativeMethods的商业决策
伦敦,汤姆森学习出版集团
休Coolican (1990)研究方法和统计inPsychology伦敦,霍德和斯托顿
斯威夫特,路易斯(2001)的定量方法商业,管理及财务,贝辛斯托克,帕尔格雷夫