The Chi-Square Test

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Analysis
Introduction
The following statistical techniques are used to find relationships between one or more variables. In this case, the study of these techniques is to find their significance, uses and limitations as an approach of investigating the financial and operating performance of firms before and after Initial Public Offerings. These statistical methods will be used to study how the performance of most firms does indeed deteriorate after they go public. The markets in question are the Saudi Arabian firms where 52 public offerings made by the Gulf Cooperation Council will be investigated.
The Chi-Square Test
The Chi-Square test determines the difference between observed frequencies and expected frequencies in one or more categories. Questions like whether the number of objects or individuals that fall in each of these categories differ significantly from the results you expected arise. The difference in results can either be due to sampling errors or it could be the real difference. There are requirements that are needed in order to perform a Chi-Square test. These requirements are the presence of one or more categories, an adequate sample size, a random sample, data recorded in frequency form, quantitative data, and independent observations (Carver & Nash, 2011). The use of prior knowledge about the frequencies observed in a previous test is used to estimate an expected frequency in the current one. The expected frequency is calculated by multiplying the sample size by each of the frequency proportions observed in the previous test. In the investigation of long run post IPO performance in Saudi Arabia, an extensive investigation of data collected between 2003 and 2010 was carried out. Al-Hassan, Delgado and Omran, after investigating stock returns of 47 IPOs were able to predict the reasons for bad performance in the long run. The limitations of the chi-square method in delicate cases like these, is that to confirm normality, it has to be used alongside another different technique like the t-test method. Another limitation of the chi-square is that all values measured must be independent. This means that if you fit an extra category into a chi-square, the analysis is no longer appropriate. The other limitation is that all the data in a chi-square must be frequency data. For example, if you are calculating how many firms perform better after an IPO versus how many firms perform much worse, then a chi square is appropriate. The chi square technique is also dependent and sensitive to the sample size (Carver & Nash, 2011). The chi-square technique is not appropriate for small sample sizes. The chi square does technique does not give so much information about the strength of the relationship between two variables.
Comparing means: The T-test and ANOVA
The T-test analysis is used to compare group means. It is normally used to determine differences in two sets of averages (Mitchell & Jolley 2010). The t-test is limited to only comparing two groups. One way ANOVA on the other hand can be used to compare more than two groups. The t-test assumes that samples selected for comparison come from normally distributed populations with unknown averages. Using the ma...