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

Summary
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…

## Extract of sample "#2 (Z and T) (Megastat/Minitab)"

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.
Abstract 2
Introduction 4
Z – Test 4
Hypothesis Testing 4
Student’s t-test 5
Hypothesis testing 5
Bibliography 6
Gujarati, Damodar N. 2003. Basic Econometrics. McGraw-Hill Publishers: New York. 6
Appendix 7
For Z-test 7
For Student’s t-test 7
Introduction
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 Testing
1. 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-test
In 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 testing
1. 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.
Bibliography
Gujarati, Damodar N. 2003. Basic Econometrics. McGraw-Hill Publishers: New York.
Appendix
For Z-test
1
Alabama
521.76
38
Oregon
376.81
3
Arizona
722.21
39
Pennsylvania
1234.83
4
Arkansas
328.19
40
Rhode Island
107.04
6
540.7
41
South Carolina
504.26
7
Connecticut
404.17
17
Kansas
321.26
8
Delaware
83.6
18
Kentucky
485.8
9
District of Columbia
57.11
20
Maine
136.28
11
Georgia
1118.38
22
Massachusetts
682.18
12
Hawaii
128.79
23
Michigan
1211.44
14
Illinois
1483.65
26
Missouri
628.67
27
Montana
98.67
44
Texas
3184.24
28
194.82
49
West Virginia
197.56
30
New Hampshire
140.24
50
Wisconsin
577.95
32
New Mexico
227.9
51
Wyoming
57.29
33
New York
1942.96
47
Virginia
839.69
36
Ohio
1267.09
48
Washington
695.41
For Student’s t-test
3
Arizona
722.21
33
New York
1942.96
5
California
4507.92
35
North Dakota
67.12
7
Connecticut
404.17
36
Ohio
1267.09
8
Delaware
83.6
37
Oklahoma
452.95
9
District of Columbia
57.11
17
Kansas
321.26
13
Idaho
178.22
22
Massachusetts
682.18
14
Illinois
1483.65
23
Michigan
1211.44
27
Montana
98.67
44
Texas
3184.24
28
194.82
46
Vermont
65.94
30
New Hampshire
140.24
49
West Virginia
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