Statistical Analysis of the T-test - Book Report/Review Example

 Statistical Analysis of the T-test Introduction T-test is a tool that is used in statistical analysis to examine two population means. A two sample t test can be used to examine whether there are differences between two samples especially when a small sample size is used or when there is unknown variances within the two sample distributions (Rasch, 2011). It has an advantage in that you are not required to have knowledge of the deviations particularly the standard deviation of the population. It can therefore be used especially when testing hypothesis of a population that is unknown completely with respect to population mean (µ) as being unknown as well as the experimental error and information available on the population is from a sample. According to (Reed & Stark, 2004) t test evaluates statistically, the mean of two populations and examines if the taken samples are significantly different and more so is used if variances are unknown and involves a small sample size. Statistical Analysis of the t test The t-test on the hypothesis requires the mean of the population and a sample. There are certain situations when this test on the hypothesis is used for example when there is need for a researcher to determine whether treatment does bring a change especially on the mean of the population (Cao, 2007). From the population, a sample is usually obtained and then a treatment is administered to that sample. A conclusion is made if the mean of the sample has a significant difference from the original one. Secondly, t-test can be used also when a predicting or when theory provides a value of hypothesized population means whose actual mean is unknown (Reed & Stark, 2004). If the difference is significant, the indication is that the value hypothesized for population mean (µ) be rejected. T-test is very important in statistical analysis of discrepancies in sampling. In sampling statistics, there is possibility of discrepancy usually between the mean of the population and that of the sample and this phenomenon is referred to as sampling error. The main goal of a t-test on the hypothesis is usually to find and evaluate the significance in discrepancy (Reed & Stark, 2004). Statistical analysis of the T-test outlines a critical step in testing hypothesis as described above. T-test involves calculation of the exact differences in population mean (µ) and sample mean (M) and whether it is reasonably expected. However, since the standard deviation of the population is unknown it becomes impossible to calculate the standard error of sample M. As stated by Rasch et al (2011), the T-statistical test will therefore require one to have sample data in order to get the standard error estimate of the sample mean (M). The formula of the calculating the error is given as follows sM=s2/n or Sm=s/square root of n (Reed & Stark, 2004) Another importance of the statistical t- test is that a ratio is usually formed after the calculation, usually the top part of that ratio contains the difference obtained from the mean of the sample and that of the mean population hypothesized while bottom part of that ratio is usually standard error that measures what difference is to be expected through chance. T = obtained difference/standard error = M-µ/sM (Reed & Stark, 2004) When the value of t is big that is there is a big ratio, the indication is that the difference obtained between hypothesis and data is greater than the expectation had the treatment had no effect. The t-test can be thought of as a z-score that is estimated. This estimation is a resultant of using variance of the sample in order to get the variance of the unknown in population. When the samples are large there is good estimation and t-statistic appear similar to z-score and in case the sample is small the test provides a poor estimation of z-score (Reed & Stark, 2004). In statically analysis of t test there is value of degree of freedom (df) which is stipulated as shown below. This is a useful description of how the t statistic will represent something known as z-score. df = n-1. (Reed & Stark, 2004) The value of degree freedom obtained will help in determining whether the t approximates distribution form is normal. For instance if the values of degree of freedom are large, the t- distribution is nearing normal distribution while for values that are small the distribution will appear more spread and flatter compared to the normal distribution (Cao, 2007). In evaluating the t-test usually in hypothesis testing, you select α and find out the value of degree freedom from the t-statistic. While consulting on the distribution table of t, you reject the null hypothesis when the statistics of t obtained are bigger than the table’s critical value (Reed & Stark, 2004). The test on hypothesis takes some procedures as for case in testing other scores such as z-score. The first step involves stating the hypothesis as well as selecting a value for α. It is worth noting that for the null hypothesis, the value of µ is specific. Secondly locating the region that is critical in this case therefore you get the value of degree of freedom and also use the distribution table. This is followed by calculating the test statistic and finally making a decision. The decisions are either based on reject or fail to reject the null hypothesis. In measuring the effect of size using the t-test, the effect caused by the treatment introduced has some significance in that it is partially determined by the sample size as well as the size of effect. However assumptions are not made on the basis of the significance of the effect is caused by an effect that is large. Researchers therefore recommend the computation of the hypothesis be measure along that of the size of effect. Mean difference/standard deviation (Sd) =M-µ/ss (Reed & Stark, 2004) In t-test the effect of treatment introduced is measured in term of the standard deviation. In this test also it is possible to compute the percentage in variance to be accounted due to treatment. The idea behind the measure of treatment is that such treatments cause changes in score and contributes to variability observed in data (Rasch et al 2011). For hypothesis test we can measure the variability that is attributed to treatment thus obtain the extent of treatment effect. Percentage of variance accounted for = r2 = t2 t2+df (Reed & Stark, 2004) The T-test usually assess the mean of groups mostly two and whether there are different from one another statistically. It addresses the question of whether the means of the two groups of population are statistically different (Reed & Stark, 2004). In this case, the value of t is either positive or negative. For instance the value becomes positive when the second mean is smaller than the first one and turns negative if the first mean is smaller. In research there is the significance table used to test if the large ratio in difference found is not due to finding chance. In testing the significance there is a level known as risk or alpha. In social research mostly there is a set rule known as rule of thumb and alpha is set at (Cao, 2007). This means that in five out of a hundred there would be a difference which has statistical significance to the means. In statistical analysis of t-test, we can use the degree of freedom, alpha level and the t value to determine if the t value has some significance depending on whether it is large or small. For a significant value the conclusion is that mean difference of the two groups is different. Conclusions While testing mean of sample that are independent some assumption are made and applied. One is that we assume the drawn at random from their population and that there are independent. Secondly is that the two samples have measurement of scale and has an interval scale that is equal. Lastly is that the population from which the two samples are drawn is having a distribution that is normal. In carrying out the test the assumption are applied as long as they are met. However, t test provide the simpler yet effective way of analyzing tow samples and making inferences as to their variability and similarities which are important in statistics. References Cao, H. (2007). Moderate deviations for two sample t-statistics. ESAIM.Probability and Statistics, 11, 264-271. Rasch, D., Kubinger, K. D., & Moder, K. (2011). The two-sample t test: Pre-testing its assumptions does not pay off. Statistical Papers, 52(1), 219-231. Reed, James F., I., II, & Stark, D. B. (2004). Robust two-sample statistics for testing equality of means: A simulation study. Journal of Applied Statistics, 31(7), 831-854. Read More