October 02, 2012
Z Scores, Z Tests and t Tests
Overview and Review
At the beginning of the course we learned that there are two branches of statistics, namely, parametric and non-parametric. Further we learned that parametric statistical processes are broken down into two other categories, namely descriptive statistical processes and inferential. We learned also that descriptive statistics (mean, mode, median, standard deviation, and frequencies) are only to be used to describe the characteristics of the data rather than draw conclusions of make inferences from the measurement data collected. However, the importance of descriptive statistics cannot be undermined as they form the basis for the workings of inferential statistical processes – especially the mean. In data analysis one of the most important concepts to remember is that regardless of the topic or issue being investigated all is based on the mean of a data set. Although we cannot draw conclusion or make predictions from descriptive statistics their usefulness in inferential statistics is significant.
As stated inferential statistics is a branch of statistics that is used in making inferences about traits or characteristics of a greater population on the basis of sample measurement data. The primary goal of inferential statistics is to leap beyond the measurement data at hand and make inferences about a greater population. Take for example a psychologist who is interested in knowing whether a new behavior modification product will likely be a seller in a certain market area. Knowing that the entire consumer population cannot be queried as to market acceptance, the psychologist would select a representative sample for the area, administer whatever measurement instrument is necessary to garner the data and, on the basis, of the sample data results, determine whether or not the new product will be profitable. The statistic used to determine whether or not the sample is representative of the entire market population would be an inferential statistics.
When using inferential statistical processes to generate information in order to make predictions about a larger population the chosen sample must always be on the basis of random selection or random assignment. Without random sampling or random assignment the mathematical values received by way of the statistical analysis are in err. Or, another way of putting is to say that the results would be “Lies, damn lies” about the data analyzed.
For convenience purposes throughout the remainder of this course the following symbols will be used most extensively. Statisticians, regardless of area, use English letters to denote sample statistics and Greek letters to symbolize population parameters.
NameSample StatisticPopulation Parameter
MeanX µ (mu)
VarianceSD² σ 2 (sigma squared)
Standard DeviationSD σ (sigma)
Correlationr ρ (rho)
Proportionp π (pi)
Regression Coefficient b β (beta)
When trying to arrive at conclusions that extend from the measurement data alone, inferential statistics are the data analysis tools of choice. For example, inferential statistics are used to infer from the sample data to the larger population data or when there is an need to make judgments of the probability that an “observed” difference between groups is an accurate and dependable one and not those that happened by chance alone. In order too accomplish that which inferential statistics were designed two models are available: estimation testing and hypothesis testing. In the estimation model the sample measurement data is used to estimate a parameter (population) and a confidence interval about the estimate is created. The confidence interval is basically the range of values that has a high likelihood of containing the parameter. The parameter is a numerical value that measures some...