#2 (Z and T) (Megastat/Minitab) - Case Study Example

Case Study 2 wise Variation in the level of elementary education in USA The purpose of this paper is to find out how far the results derived in the previous case study tally in context of in-sample observations. Statistical tests like Z-test and Student’s t-test have been included for a reliability check in this regard. While Z-test deals with samples having a larger size, the Student’s t-test is more suitable for smaller samples thus making it possible for an intricate inspection.Table of ContentsAbstract 2Introduction 4Z – Test 4Hypothesis Testing 4Student’s t-test 5Hypothesis testing 5Bibliography 6Gujarati, Damodar N. 2003. Basic Econometrics.

McGraw-Hill Publishers: New York. 6Appendix 7For Z-test 7For Student’s t-test 7Introduction Both z and Student’s t tests are used to infer about the reliability of a sample, i.e., whether it is drawn from a particular population or not. The feature being tested in each case is the presence of any significant difference between the sample and the population means. A significant difference would suggest that the sample is not a reliable representative of the population, while the situation is reversed if there is no significant difference.

However, Z-tests are more suitable for larger samples viz., n ≥ 30 and t-tests are used when the sample size is less than 30.Z – Test From the data on the number of enrollments in elementary school in USA in the year 2004, 32 observations are chosen at random (details included in the appendix) and the sample mean is being calculated. The steps that confirm that the sample is drawn from the concerned population are described as follows.Hypothesis Testing1. The sample mean had been found out to be M = 640.65. Hence the objective is to test the null hypothesis H0: μ = 640.

65, against the alternative hypothesis, H1: μ ≠ 640.65.2. Let the critical region be 5%, i.e., the level of significance to be used here is 0.05.3. Now, the population mean, μ has been found out as 670.2, and the SD, σ = 800.98. Hence, Standard Error = σ / √n = 141.59.Thus, estimated z statistic = (μ – M) / σ / √n = 0.2087.4. The probability value or the percentage of times that the above calculated z-statistic occurs = Φ (0.2087) = 0.08 = 8%.5. The probability value is greater than the level of significance implying that the null hypothesis being considered cannot be rejected at 5% level of significance.

Student’s t-testIn this case, 20 random observations are being considered (shown in the appendix). The sample mean, M = 863.1675. The population mean and standard deviations have been found to be 670.2 and 800.98 respectively, implying that the standard error, σ / √n = 179.10.Hypothesis testing1. Here the objective is to test the null hypothesis H0: μ = 863.1675, against the alternative hypothesis, H1: μ ≠ 863.1675.2. Let the critical region be 5%, i.e., the level of significance to be used here is 0.05.3.

The estimated Student’s t statistic is calculated in the same way as the z-statistic; the only difference lies in the sample size. Hence, the calculated t-statistic = (μ – M) / σ / √n = 1.077.4. The probability value or the percentage of times that the above calculated Student’s t-statistic occurs = Φ (1.077) = 0.3577 = 35.77%5. The probability value is much higher than the level of significance implying that the null hypothesis being considered cannot be rejected at 5% level of significance.

BibliographyGujarati, Damodar N. 2003. Basic Econometrics. McGraw-Hill Publishers: New York.AppendixFor Z-test1Alabama521.7638Oregon376.813Arizona722.2139Pennsylvania1234.834Arkansas328.1940Rhode Island107.046Colorado540.741South Carolina504.267Connecticut404.1717Kansas321.268Delaware83.618Kentucky485.89District of Columbia57.1120Maine136.2811Georgia1118.3822Massachusetts682.1812Hawaii128.7923Michigan1211.4414Illinois1483.6526Missouri628.6727Montana98.6744Texas3184.2428Nebraska194.8249West Virginia197.

5630New Hampshire140.2450Wisconsin577.9532New Mexico227.951Wyoming57.2933New York1942.9647Virginia839.6936Ohio1267.0948Washington695.41For Student’s t-test3Arizona722.2133New York1942.965California4507.9235North Dakota67.127Connecticut404.1736Ohio1267.098Delaware83.637Oklahoma452.959District of Columbia57.1117Kansas321.2613Idaho178.2222Massachusetts682.1814Illinois1483.6523Michigan1211.4427Montana98.6744Texas3184.2428Nebraska194.8246Vermont65.9430New Hampshire140.2449West Virginia197.56