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Hypothesis Testing
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Abstract
Hypothesis testing refers to the scientific technique of analyzing ideas and claims on a parameter of interests using a particular population. Usually, researchers use the information provided to establish the claim. Regarding the calculations, they are used to provide a solid ground for making decisions concerning the general effects of the factor on the sample. The banking industry uses the Return on Equity ratio to determine the financial performance of the entities. This paper uses a 20 banking institutions in calculating ROE prior to and after the introduction of the Sarbanes-Oxley Act. The outcome of the calculation is used to determine the ROE of the banks. Additionally, null and alternative hypothesis are stated while at the same time, the various levels f conducting hypothesis are selected. Lastly, the paper analyzes the type 1, and two error stating the type tat occurred and reasons behind their occurrence.
Introduction
Hypothesis testing is a crucial aspect of statistics. Normally, it helps in determining the most correct and accurate statement about a sample by analyzing mutually exclusive statements. In the case of the banking sector, it uses the ROE ratio for performance evaluation (Khadafi et al., 2014). Return on Equity is the ratio of the income to the stakeholders' Equity in a given year.
ROE=Annual net income /Average stakeholders Equity
Net income is the net profit after tax. The average of the stakeholders is calculated by adding stakeholders at the start and end of the year, then divide by two. While the former is from the income statement of the entity, the latter can also be attained from the stability sheet (Kijewska, 2017). Several banks are samples in the paper to help calculate ROE. The years considered include 2001, 2002, and 2003. The period coincides with the time Sarbanes-Oxley (sox) act of 2002 was enacted.
The Act is also known as the Public Company Accounting reforms and Investors Protection Act. The improvement was generated to assist in protecting stakeholders from duplicitous financial reporting by firm. It was meant to be a strict reform to prevailing securities guidelines and executed severe new penalties on lawbreakers. It was a response to the financial scandals in the early 2000s involving companies that are traded publicly.
Calculations
ROE was calculated using the formula
ROE=Annual net income average stakeholders Equity
For example, in the case of BAC Florida Bank in the years ended financial earned 31st December 2003, the net income was $ 174,775 while the stakeholder's Equity on 31st December 2002 and 31st December 2003 was $ 2,868,126 and $ 3,042,901 correspondingly.
Using the method; ROE=Annual net income average stakeholders Equity
ROE= 174,775 / {(2,868,126 +3,042,901)/2}
=174,775 /2,955,513 =0.059 =5.9%
ROE is used in comparison of performance of the corporations in an industry to determine their performance. It is the extent of the ability of the management to produce income from the Equity available to it. Usually, a good ROE range is between 15-20%.
Bank (Financial year)
Stakeholder Equity at the start of the financial year.
Stakeholder Equity at the end of the financial year
Net Income($)
ROE (%)
1
BAC Florida (2000/2001)
$ 9,266,100
$ 2,759,678
$ 92,578
92578/ {(9266100+2759678) / 2} = 62578/6012889
= 0.015
= 1.5%
BAC Florida (2002/2003)
$,868 126
$ 3,042 901
$ 174,775
174,775 / {(2,868,126 +3,042,901) / 2} = 174,775 /2,955,513
= 5.9
2
US bank (2000/2001)
$ 15, 168
$ 16 461
$ 1,706.5
1,706.5/ {(16 461 +15, 168)/ 2}
= 1,706.5/15,814.5
= 0.1079
= 10.79
US bank (2002/2003)
$ 19,393
$ 17,273
$ 3,732.6
3,732.6 / {(17,273+19,393)/2}
=3,732.6/18,333
= 0.2036
= 20.36
3
Jp Morgan chase & co. (2000/2001
$ 20,226
$ 22,440
$ 2,638
2,638 / {(22,440+20,226)/2} = 0.1237
= 12.37
Jp Morgan chase & co (2002/2003)
$ 22,440
$ 23,419
$ 3,535
3,535 / {(3,535+323,419)/2} = 0.1541
= 15.41
4
Federal Financing (2002/2003)
$ 44,177,413
$ 40,649,720
$ 357,335
357,335 / {(40,649,720 + 44,177,413)/2}
= 0.0084
= 0.84
Federal Financing (2002/2003)
$ 40,096,800
$ 35,911,233
$ 856,618
856,618 / {(35,911,233 + 40,096,800)/2} = 0.0225
= 2.25
5
Farm Credit Bank of Texas (2001/2002)
$ 326,000
$ 369,000
$ 24,878
24,878 / {(369,000 + 326,000)/2}
= 0.0716
= 7.16
Farm credit Bank (2002/2003)
$ 369,000
$ 478,000
$ 64,824
64,824 / {(478,000 + 369,000)/2}
= 0.1530
= 15.30
6
Citigroup(2000/2001)
$ 66,206
$ 81,247
$ 14,126
14,126 / {(81,247 + 66,206)/2}
= 0.1916
= 19.16
Citigroup (2002/2003)
$ 92,900
$ 104, 100
$ 9,600
9,600 / {(104, 100 + 92,900)/2}
= 0.0970
= 9.70
7
Anguilla National Bank (2000/2001)
64, 728
77, 062
8317
8317 / {(77, 062 + 64, 728)/2}
= 0.1173
= 11.73
Anguilla National Bank (2002/2003)
$ 127,459
$ 144,836
$ 7839
7839 / {(144,836 + 127,459)/2}
= 0.0580
= 5.80
8
ECHMB (2000/2001)
$ 11,556.104
$ 11,556.104
$ 1,143.147
1,143.147 / {(11,556.104 + 11,556.104)/2}
= 0.0989
= 9.89
ECHMB (2002/2003)
$ 33,172.531
$ 38,957.633
$ 1,484.276
1,484.276 / {(38,957.633 + 33,172.531)/2}
= 0.1883
= 18.83
9
Bank of America (2000/2001
$ 47,556
$ 48,455
$ 6792
6792 / {(48,455 + 47,556)/2}
= 0.1414
= 14,14
Bank of America (2002/2003
$ 54,567
$ 67,453
$ 7543
7543 / {(67,453 + 54,567)/2}
= 0.1236
= 12.36
10
Morgan Stanley (2000/200)
$ 19,271
$ 20,108
$ 3,521
3,521 / {(20,108 + 19,271)/2} = 0.1788
= 17.88
Morgan Stanley (2002/2003)
$ 21,885
$ 24,867
$ 3,787
3,787 / {(24,867 + 21,885)/2}
= 0.0622
= 6.2
11
Wells Fargo (2000/2001)
$ 27,214
$ 30,358
$ 5,710
5,710 / {(30,358 + 27,214)/2}
= 0.1866
= 18.66
Wells Fargo (2001/2002)
$ 30,319
$ 34,469
$ 6,202
6,202 / {(34,469 + 30,319)/2}
= 0.1936
= 19.36
12
Goldman Sachs (2000/2001)
$ 16,530
$ 18,231
$ 3,067
3,067 / {(18,231 + 16,530)/2}
= 0.1764
= 17.64
Goldman (2002/2003)
$ 35,557
$ 40,379
$ 3,005
3,005 / {(40,379 + 35,557)/2} = 0.0079
= 7.9
13
Bank of New York Mellon (2000/2001)
$ 3,482
$ 3,395
$ 1,318
1,318 / {(3,395 + 3,482)/2} = 0.0035
= 0.35
Bank of New York Mellon (2002/2003
$ 682
$ 701
$ 588
588 / {(701 + 682)/2}
= 0.0851
= 0.85
14
PNC financial Services (2000/2001)
$ 6,656
$ 5,823
$ 377
377 / {(5,823 + 6656)/2} = 0.0604
= 6.04
PNC financial Services (2002/2003)
$ 6,859
$ 6,645
$ 1,001
1,001 / {(6,645 + 6,859)/2} = 0.1483
= 14.83
15
HSBC holdings (2000/2001)
$ 48,072
$ 48,444
$ 4,911
4,911 / {(48,444 + 48,072)/2}
= 0.1017
= 10.17
HSBC holdings(2002/2003)
$ 55,831
$ 80,251
$ 7,231
7,231 / {(80,251 + 55,831)/2}
= 0.1063
= 10.63
16
TD Bank (2001/2002)
$ 11,505
$ 12,144
$ 1,090
1,090/ {(12,144 + 11,505)/2}
= 0.92
= 9.2
TD Bank (2002/2003)
$ 12,144
$ 11,396
$ 1,480
1,480 / {(11,396 + 12,144)/2}
= 0.1257
= 12.57
17
Sun trust Banks (2000/2001)
$ 7,501.9
$ 8,073.8
$ 1,375.5
1,375.5 / {(8,073.8+7,501.9)/2} = 0.1766
= 17.66
Sun Trust Banks (2002/2003)
$ 8,725.7
$ 9,083.0
$ 1,332.3
1,332.3 / {(9,083.0 + 8,725.7)/2}
= 0.1496
= 14.96
18
American Express (2001/2002)
$ 7,562
$ 9,471
$ 644
644 / {(9471 +7562)/2} = 0.756
= 7.56
American Express (2002/2003)
$ 13,861
$ 15,323
$ 2,987
2,987 / {(15,323+13,861)/2} = 0.2047
= 20.47
19
Citizen financial (2000/2001)
$ 3,035.07
$ 5,730.24
$ 318.527
318.527 / {(5,730.24+3,035.07)/2} = 0.7271
= 7.27
Citizen financial (2002/2003)
$ 6,544.075
$ 8,633.045
$ 1,253.739
1,253.739 / {(8,633.045+6,544.075)/2}
= 0.1652
= 16.52
20
Charles Schwab (2001/2002)
$ 1119
$ 980
$ 429
429 / {(1,119 +980)/2} = 0.4083
= 40.8
Charles Schwab (2002/2003)
$ 980
$ 1,002
$ 408
408 / {(980 +1,002)/2}
= 0.4117
= 41.17
Average ROE of the 20 companies before the Sarbanes-Oxley act of 2002 is calculating by dividing the total ROE by total number of banks.
=(1.5+10.79+12.37+0.84+7.16+19.16+11.73+9.89+14.14+17.88+18.66+17.64+0.35+6.04+10.17+9.2+17.66+7.56+7.27+40.8)/20) = 233.81/20 =11.6905%.
Average ROE after Sarbanes-Oxley act of 2002 enactment
=(5.9+20.34+15.41+2.25+15.30+9.70+5.80+18.83+12.36+16.2+19.36+7.9+0.85+14.83+10.63+12.57+14.96+20.47+16.52+41.17) =281.35/20 =14.0675%.
The value of the average ROE before the enactment of the Act was lower than after the passage. Regarding the question, the answer is NO. The value after is higher (14.07%) as compared to 11.69% which was obtained before the enactment of the Act. The passage of the Act led to a affirmative effect regarding the performance of the enterprise. The Sarbanes-Oxley act of 2002 raised the confidence level investors, which eventually led to an increase in ROE of the different entities. The confidence of level may have grown as a result of improved internal audit control systems in the various companies. The reform ensured that entities were accountable for their operations and on use of the money invested by the investors.
Null Hypothesis and Alternative hypothesis
H0: The Enactment of the Sarbanes-Oxley act of 2002 did not have effect on the effectiveness of banks and their Return on Equity (ROE).
H1: the enactment of the Sarbanes-Oxley act of 2002 has effects on the performance of the banks together with their Return on Equity (ROE).
Significant Levels for Hypothesis Test
In statistics, the significance level is critical in determining the correlation and the association between variables. It measures the strength of the evidence that needs to be available in the sample before rejecting the null hypothesis and concluding that there is statistical significance. Usually, the significance level is determined before experimenting. Researchers’ aim of experimenting is to make a conclusion based on the result of the investigation. The outcome may either lead to acceptance of the hypothesis or rejection. The three levels of significance that can be applicable in hypothesis testing include P-value, critical regions, and the confidence levels.
P-value
P-values is the probability of the researcher obtaining in the impacts observed in the sample in cases where the null hypothesis is correct. The values are determined based on the sample data, and under the assumptions, the null hypothesis is correct. Lower p-values represent more considerable evidence against the null hypothesis, while higher values indicate little proof (Nahm, 2015). Typically, p-values identify the hypothesis supported by the sample data. P-values are always equated to the significance level before making decisions. For instance, when the p-value is greater than the level of significance, the null hypothesis should be rejected a conclusion made on there is statistical significance. Decisions are established primarily based on the evidence of the samples. Thus, it is stronger to cause rejection of the null hypothesis at the population level.
Confidence Level
The confidence level is the possibility of the parameters lying within a particular range of values. It is highly associated with the level of significance. The various levels of significance, together with confidence, include the 0.10 level of significance and a 90% confidence level (Greenland et al., 2916). The level of confidence of 0.05 is interrelated to the 95% confidence level. Last, 0,02 level of importance is associated with a 99% level of confidence. Usually, the rule for rejecting the null hypothesis states that when the p-value is low or equal to the significance level, the null hypothesis should be rejected. However, in instances where the p-value is more than the level of significance, the null hypothesis should not be dismissed.
Critical regions
The critical region is a region of values that corresponds to the rejection of the null hypothesis at particular chooses probability levels. Usually, the shaded area represents the t-distribution curve, which is equal to the level of significance. Typically, the critical values are tabulated, thus obtained from the mathematical student tables. The statistical test either uses a one-tailed or two-tailed test (Nahm, 2017). The type and number of tails are determined by the nature of the null and alternative hypothesis. The critical points are determined either by the t or z test. The essential regions are responsible for determining the level of deviation that is needed for a null hypothesis to be rejected. They are also referred to as the level alpha.
Type 1 error
Both the significance level and the p-value are essential in ensuring the control of the type 1 error in a theoretical test. A type one error happens in instances when a null that is observed to be accurate but still the researcher rejects it (Kaur & Stoltzfus, 2017). In this case, the null hypothesis take up that there were no effects of the Sarbanes-Oxley act on the performance of the banks together with the value of ROE. After an evaluation, it is realized that the Sarbanes-Oxley Act has a significant effects on the banking sector, thus a considerable deviation. In this case, there is a possibility of the occurrence of the type 1 error. Tre error can cause severe and fatal effects on the investors who would assume that the Sarbanes-Oxley act did change the banking sector.
Type 2 error
This type 2 error takes place in instances when a null hypothesis that is false fails to be rejected. The likelihood of the mistake occurring is dependent on the ability of the tests and their power. In the case of the explanation, there are no chances for the null hypothesis to be false and eventually be rejected. In the case of incidence, some banks in the industry and under similar law experienced a decline in ROE. Nonetheless, type 2 error is not as dangerous as type 1.
Conclusion
During an experiment, the hypothesis is tested to ascertain the truth or falseness of the assumptions. A researcher can choose to use the various test, including t-tests as well as the z-test. The level of confidence, together with significance level, operate and work together to give a decision wither to reject or accept a null hypothesis.
References
Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations. European journal of epidemiology, 31(4), 337-350.
Kaur, P., & Stoltzfus, J. (2017). Type I, II, and III statistical errors: A brief overview. International Journal of Academic Medicine, 3(2), 268.
Khadafi, M., Heikal, M., & Ummah, A. (2014). Influence analysis of return on assets (ROA), return on equity (ROE), net profit margin (NPM), debt to equity ratio (DER), and current ratio (CR), against corporate profit growth in automotive in Indonesia Stock Exchange. International Journal of Academic Research in Business and Social Sciences, 4(12).
Kijewska, A. (2016). Determinants of the return on equity ratio (ROE) on the example of companies from metallurgy and mining sector in Poland. Metalurgija, 55(2), 285-288.
Kim, H. Y. (2015). Statistical notes for clinical researchers: Type I and type II errors in statistical decision. Restorative dentistry & endodontics, 40(3), 249-252.
Nahm, F. S. (2017). What the P values really tell us. The Korean journal of pain, 30(4), 241.
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