In: Statistics and Probability
Scenario/Summary
A confidence interval is a defined range of values such that there is a specified probability that the value of a parameter lies within the interval.
In Part I of this project, you will pick a topic, complete research and provide a write-up that includes calculations. Round all values to two decimalplaces when appropriate.
Deliverables
Examples of Topics
Open a Word Document
a. For our analysis we have used the data for High School Graduation Rates by State in the USA. Our aim for the study is to analyse the graduation rates in each of the state in the US for a given year in order to obtain a probable range of graduation rates as per the specified confidence interval.
The data was extracted from https://worldpopulationreview.com/states/high-school-graduation-rates-by-state and was originally surveyed in the session 2016-17.
b.Following are the tables as per the specification for the dataset:
Table.1
State | Percentage |
Alabama | 89.3 |
Alaska | 78.2 |
Arizona | 78 |
Arkansas | 88 |
California | 82.7 |
Colorado | 79.1 |
Connecticut | 87.9 |
Delaware | 86.9 |
Florida | 82.3 |
Georgia | 80.6 |
Table.2
State | Percentage |
Hawaii | 82.7 |
Idaho | 79.7 |
Illinois | 87 |
Indiana | 83.8 |
Iowa | 91 |
Kansas | 86.5 |
Kentucky | 89.7 |
Louisiana | 78.1 |
Maine | 86.9 |
Maryland | 87.7 |
Table.3
State | Percentage |
Massachusetts | 88.3 |
Michigan | 80.2 |
Minnesota | 82.7 |
Mississippi | 83 |
Missouri | 88.3 |
Montana | 85.8 |
Nebraska | 89.1 |
Nevada | 80.9 |
New Hampshire | 88.9 |
New Jersey | 90.5 |
Table.4
State | Percentage |
New Mexico | 71.1 |
New York | 81.8 |
North Carolina | 86.6 |
North Dakota | 87.2 |
Ohio | 84.2 |
Oklahoma | 82.6 |
Oregon | 76.7 |
Pennsylvania | 86.6 |
Rhode Island | 84.1 |
South Carolina | 83.6 |
Table.5
State | Percentage |
South Dakota | 83.7 |
Tennessee | 89.8 |
Texas | 89.7 |
Utah | 86 |
Vermont | 89.1 |
Virginia | 86.9 |
Washington | 79.4 |
West Virginia | 89.4 |
Wisconsin | 88.6 |
Wyoming | 86.2 |
c. The mean of the sample is calculated by using the formula:
where, sum of terms = 4237.10
and, number of terms = 50
so, mean, x = 84.74
The standard devition of the sample is calculated by using the formula:
so, standard deviation, s = 4.32
also, √n = 7.07
The following formula is used for calculating confidence interval:
where xbar is the mean
and z is the z score for the given margin of error
1- 80% confidence interval:
CI80% = 84.74 +- 1.28 * (4.32/7.07) = 84.74 +- 0.78 = (85.52, 83.96)
Here, margin of error is 20%
2- 95% confidence interval:
CI95% = 84.74 +- 1.96 * (4.32/7.07) = 84.74 +- 1.19= (85.93, 83.55)
Here, margin of error is 5%
3- 99% confidence interval:
CI99% = 84.74 +- 2.58 * (4.32/7.07) = 84.74 +- 1.58= (86.32, 83.16)
Here, margin of error is 1%
4- 98% confidence interval:
CI98% = 84.74 +- 2.33 * (4.32/7.07) = 84.74 +- 1.42= (86.16, 83.32)
Here, margin of error is 2%
c. The average rate of graduation in 50 states of USA is 84.74 percentage with a standard deviation of 4.32 percentage. The confidence interval we calculated earlier would help us evaluate the range of probablity of a student graduating from any randomly picked-up state of the USA with a specific margin of error which can have various useful information ranging from the census to the effects of evaluation of any changes brought about in the education system by the government.
The confidence interval tends to broaden as the confidence level rises and vice versa. This happens due to the fact that as the confidence level rises, we tend to be less confident about our estimetes on a specific number and hence the range of values we want to have confidence increase. So, mathematically the Z-score is acts like the varuiable thats starts to increase as the margin of error deminishes.
1- 80% confidence interval:
For 80% confidence interval the graduation rates are expected to be in the range 85.52% to 83.96%
Here, margin of error is 20%
2- 95% confidence interval:
For 95% confidence interval the graduation rates are expected to be in the range 85.93% to 83.55%
Here, margin of error is 5%
3- 99% confidence interval:
For 99% confidence interval the graduation rates are expected to be in the range 86.32% to 83.16%
Here, margin of error is 1%
4- 98% confidence interval:
For 98% confidence interval the graduation rates are expected to be in the range 86.16% 83.32%