Unit 3 -IP- Samples - Essay Example

Hypothesis Testing: Unit 3 – IP Samples The aim of this paper was to perform a two – tailed t – test on both, the legal services satisfaction and the sentence satisfaction, using a 0.05 significance level. The t – test used was a two – sample t – test. The test failed to reject the null hypothesis that the average difference between the two variables is the same. Descriptive results indicate that the means of the two variables are similar. In total, there are twenty five samples of each variable.

Their means are similar: the mean of the legal services satisfaction is 5.232, whereas the mean of the sentence satisfaction is 4.952. In order to test whether the two variables have the same means, a two – sample, two – sided t – test must be performed. The two - tailed test is used because the concern is not whether some variable has a larger or lower mean, but whether the two means are different. As a result, the null hypothesis was established in equation (1) below, stating that the difference in means of the two samples is random. (1)Likewise, the alternative hypothesis is that the two sample means are not the same, as can be seen in equation (2) below:(2)Since the sample sizes are relatively small, the t – test is used as the preferred test (Baltagi, 2008, p.31). The confidence interval is constructed in order to test this hypothesis.

The confidence set with confidence level must be determined such that the true difference of the sample means is included in this set. What this implies is that the confidence set will include the true difference of population means with probability “at least ” (Paolella, 2011, p. 13), as shown in equation (3) below:(3)The confidence level chosen is 0.95, which implies that the corresponding confidence set will contain the true population mean difference with 95% probability (Paolella, 2011, p.13). The probability of rejecting the null hypothesis when it is true is as a result 5% (Baltagi, 2008, p.21). This error is denoted as type I error.

The larger the confidence interval, the larger the probability that the null hypothesis will be accepted when it is in fact false (Baltagi, 2008, p.22). This error is denoted as type II error. A t – score must be calculated by subtracting the sample means and dividing by the square root of the adjusted population variances, which can be estimated through sample variances. There are in total degree of freedom (Neustadl, n.d.). The resulting t – value is ± 2.009, which is also the critical region (Dawson, 2012;Ruud, 2000, p.224). The t – test can be seen in equation (4) below:(4)The t – test is 0.

929, which is within the ± 2.009 interval. As a result, the null hypothesis cannot be rejected (see Tables 1 and 2 in the Appendix). The test conducted was chosen because of a small sample size. When there is a small sample size, the confidence interval cannot be estimated correctly with the z – test (Baltagi, 2008, p.31). The t – test is preferred because of the data dependence. Both variables have in common the type of offense, meaning that variables are not independent. As a result, the t – test is preferred (Gaten, 2000).

Samples have been used for practical purposes. First off, populations are usually large, which makes computing hard. Secondly, all members of the population are hard to locate and collect necessary data from. As shown, t – test is used for small samples. In this example, the null hypothesis could not be rejected, indicating the difference in means is random. ReferencesBaltagi, B.H. (2008). Econometrics. Berlin: Springer – Verlag. Dawson, R.J.M. (2012). Table of t values. University of Missouri.

Retrieved fromhhttp://oak.snr.missouri.edu/nr3110/pdf/table1.pdf Neustadl, A. (n.d.). Difference of Two - Means Test. University of Maryland. Retrieved from http://www.bsos.umd.edu/socy/alan/handouts/diff_of_two_means.pdf Gaten, T. (2000). Z – tests and t – tests. Leeds University. Retrieved from http://www.le.ac.uk/bl/gat/virtualfc/Stats/ttest.html Paolella, M. (2011). Fundamental Statistics: A Computational Approach. University of Zurich: EPub. Ruud, P.A. (2000).An Introduction to Classical Econometric Theory.

Oxford: Oxford University Press.AppendixTable 1DataLegal ServicesSentence Satisfaction6,25,54,23,66,23,84,35,63,25,54,54,44,73,15,36,94,94,63,46,85,55,55,83,86,23,24,24,65,76,56,46,13,73,96,24,66,55,55,14,46,24,76,36,26,34,75,25,54,64,8Table 2The t – test ResultSum130,8123,8Mean5,2324,952Variance1,03061,1509333Mean difference0,28Denominator0,30149t - test0,92872