Descriptive Statistics and Hypothesis Testing: Everything You Need to Know

High school graduation is among the first and most important milestones in the life of young scholars. Unfortunately, a few scholars fail to graduate from high school. There are several reasons why a student may fail to graduate -from being homeless, coming from an economically disadvantaged family, or even having a less-than-ideal home life. Due to these and many other reasons, the high school graduation rates in the United States is not 100%. In addition, the rates vary across states as well as across student demographic groups. In this paper, sample data of the high school graduation rates by state in 2020 will be collected and analyzed. The analysis will include the calculation of descriptive statistics, such as the mean and the standard deviation, and confidence intervals (CI). 

The sample data for high school graduation rates by states were obtained from the World Population Review website, a website that collects demographic data on the population of countries and cities. The data is available at https://worldpopulationreview.com/state-rankings/high-school-graduation-rates-by-state . Table 1 shows the data that was retrieved from this website. 

Table 1: High School Graduation Rates by State: 2020 

93.20	91.90	90.40	88.60	86.50
93.00	91.80	90.40	88.00	86.30
92.90	91.70	90.20	88.00	86.20
92.90	91.40	90.10	87.80	85.80
92.70	91.10	90.00	87.40	85.70
92.60	91.10	89.80	87.10	85.30
92.50	90.70	89.60	87.00	84.80
92.30	90.60	89.50	86.80	83.90
92.00	90.50	89.30	86.70	83.20
92.00	90.50	88.90	86.50	82.90

Source: World Population Review (2020). 

Using the sample data collected, the sample mean and standard deviation were calculated using Excel function. Table 2 shows the results obtained. 

Table 2: Mean and Standard Deviation 

Mean	89.20
Standard Deviation	2.7844

The mean is the average of the sample data. In Excel, the "AVERAGE" function is used to calculate the mean. Standard deviation is a measure of the amount of variation between values in a given dataset. The mean and standard deviation of the collected sample data is 89.20% and 2.7844, respectively. Other than the descriptive statistics, 80%, 95%, and 99% CIs was calculated using the sample data. The formula for calculating CI is shown below: 

Where, 

80% CI 

For 80% CI, 

Upper Level 

Lower Level 

Thus, 

Margin of Error 

95% CI 

Upper Level 

Lower Level 

Thus, 

Margin of Error 

99% CI 

Upper Level 

Lower Level 

Thus, 

Margin of Error 

My Own CI 

98% CI 

Upper Level 

Lower Level 

Thus, 

Margin of Error 

Analysis and Reflection 

As the confidence level rises, the margin of error increases as well. The error margin is influenced by three parameters: the sample size, the standard deviation, and the confidence level. In our case, the sample size and the standard deviation remained constant, whereas the confidence level was changed. This means that as the confidence level increases, the margin of error increases and vice versa. The margin of error increases because the critical value increases as the confidence level increases. 

For 80% CI, the upper limit and lower limit was found to be 89.7060 and 88.6980, respectively. This means that we are 80% confident that states' high school graduation rates in 2020 lie between 89.7060 and 88.6980. For 95% CI, the upper limit and lower limit was found to be 89.9738 and 88.4302, respectively. This means that we are 95% confident that states' high school graduation rates in 2020 lie between 89.9738 and 88.4302. For 99% CI, the upper limit and lower limit was found to be 90.2179 and 88.1861, respectively. This means that we are 99% confident that states' high school graduation rates in 2020 lie between 90.2179 and 88.1861. Lastly, for 98% CI, the upper limit and lower limit was found to be 90.1195 and 88.2845, respectively. This means that we are 98% confident that states' high school graduation rates in 2020 lie between 90.1195 and 88.2845. 

Part I of the statistics project has helped me learn how to calculate and interpret CIs. I can calculate the CI for any interval level as long as the sample data or sample parameters are provided. The project has also helped me learn how to determine the margin of error as well as determine the relationship between the margin of error and confidence intervals. Overall, this project has helped me understand the concept of confidence intervals better. 

Statistics Project #2: Hypothesis Testing 

Hypothesis testing is a method in statistics that involves testing assumptions on a given population parameter. Using sample data, one can assess the plausibility of a hypothesis through a hypothesis test. In this paper, hypothesis testing will be used to determine if a given claim is true. Sample data that pertains to birth, death, marriages, and divorces will be retrieved from the Centers for Diseases Control and Prevention (CDC), a public health institute whose main aim is to protect public health safety. 

The data that was retrieved from the CDC was collected by Rate N and published in 2009 in the National Vital Statistics Reports. The report is composed of a wide range of data sets. However, only the data sets for 2009 will be used in this paper for analysis. Table 1 shows this data. 

Table 1: Births, Deaths, Marriages, and Divorce by State, 2009 

State	Live Births	Deaths	Marriages	Divorces
Alabama	5,352	4,330	2,684	1,651
Alaska	861	286	361	381
Arizona	7,775	4,026	3,236	1,916
Arkansas	3,400	2,590	2,489	1,355
California	45,831	21,135	15,208	- - ­
Colorado	5,572	2,824	1,316	1,901
Connecticut	3,060	2,477	2,046	1,016
Delaware	891	658	321	231
District of Columbia	639	338	104	115
Florida	18,622	14,624	10,002	6,055
Georgia	11,884	6,039	4,250	- - ­
Hawaii	1,573	774	4,250	- - ­
Idaho	1,849	999	820	635
Illinois	14,166	10,031	5,129	2,740
Indiana	7,112	4,993	4,923	- - ­
Iowa	3,247	2,513	796	530
Kansas	3,469	2,104	1,683	1,027
Kentucky	4,742	2,996	1,714	1,634
Louisiana	5,659	3,078	624	- - ­
Maine	1,052	1,256	612	279
Maryland	6,411	3,804	2,170	1,288
Massachusetts	6,010	4,576	2,075	1,274
Michigan	9,206	7,585	2,895	3,006
Minnesota	5,765	3,335	1,333	- - ­
Mississippi	3,703	2,406	865	821
Missouri	6,472	4,820	2,306	1,814
Montana	936	761	398	399
Nebraska	2,190	1,304	306	153
Nevada	3,292	1,761	7,416	1,531
New Hampshire	1,031	846	451	279
New Jersey	8,946	6,126	2,349	1,957
New Mexico	2,292	1,259	952	626
New York	21,072	13,221	8,442	4,171
North Carolina	10,492	6,916	4,448	2,895
North Dakota	741	505	220	71
Ohio	11,691	9,577	4,067	2,312
Oklahoma	4,802	3,165	1,987	1,475
Oregon	3,663	2,779	1,433	1,067
Pennsylvania	11,991	11,397	3,721	3,138
Rhode Island	933	802	337	222
South Carolina	5,086	3,615	1,835	939
South Dakota	990	617	314	242
Tennessee	6,909	5,312	5,729	2,148
Texas	34,363	14,469	11,867	2,014
Utah	4,223	1,243	1,743	1,849
Vermont	458	450	277	279
Virginia	8,697	5,245	3,839	2,793
Washington	6,937	4,054	2,711	2,187
West Virginia	1,768	1,907	747	736
Wisconsin	5,415	3,784	1,184	1,525
Wyoming	605	352	81	79
Puerto Rico	3,226	1,680	2,450	1,661

Source: Rate N (2009). 

Preliminary Calculations 

The preliminary calculations calculated include the mean, median, sample standard deviation, and minimum and maximum values for each of the data sets. The results are summarized in Table 2, 3, 4, and 5. 

Table 2: Summary Table for Live Births 

Summary Table for Live Births
Mean	6,674
Median	4,772
Standard Deviation	8,106
Minimum	458
Maximum	45,831

Table 3: Summary Table for Deaths 

Summary Table for Deaths
Mean	4,187
Median	2,910
Standard Deviation	4,293
Minimum	286
Maximum	21,135

Table 4: Summary Table for Deaths 

Summary Table for Marriages
Mean	2,760
Median	1,911
Standard Deviation	3,053
Minimum	81
Maximum	15,208

Table 5L Summary Table for Divorces 

Summary Table for Divorces
Mean	1,444
Median	1,322
Standard Deviation	1,185
Minimum	71
Maximum	6,055

Hypothesis Testing 

Using the sample data and preliminary calculations, a number of hypothesis tests were performed, which are as follows: 

Determine if there is sufficient evidence to conclude the average amount of births is over 5000 in the United States and territories at the 0.05 level of significance. 

Since, 

And 

The result is not significant at p<0.05. So, we accept the null hypothesis and conclude that there is insufficient evidence to conclude the average number of births is over 5000 in the United States and territories at the 0.05 level of significance. 

Determine if there is sufficient evidence to conclude the average amount of deaths is equal to 6000 in the United States and territories at the 0.10 level of significance. 

Since, 

And 

The result is not significant at p<0.10. So, we accept the null hypothesis and conclude that there is not sufficient evidence to conclude that the average amount of deaths is equal to 6000 in the United States and territories at the 0.10 level of significance 

Determine if there is sufficient evidence to conclude the average amount of marriages is greater or equal to 2500 in the United States and territories at the .05 level of significance. 

Since, 

And 

The result is not significant at p<0.10. So, we accept the null hypothesis and conclude that there is not sufficient evidence to conclude that the average amount of marriages is greater or equal to 2500 in the United States and territories at the .05 level of significance 

Determine if there is sufficient evidence to conclude the average amount of divorces is less than or equal to 4000 in the United States and territories at the 0.10 level of significance. 

Since, 

And 

The result is not significant at p<0.10. So, we accept the null hypothesis and conclude that there is insufficient evidence to conclude that the average amount of divorces is less than or equal to 4000 in the United States and territories at the 0.10 level of significance. 

References 

Rate, N. R. N. R. N. (2009). National Vital Statistics Reports.  National Vital Statistics Reports ,  57 (13). 

World Population Review. (2020). High school graduation rates by state 2020. https://worldpopulationreview.com/state-rankings/high-school-graduation-rates-by-state