Question

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.

pick a topic, complete research and provide a write-up that includes calculations. Round all values to two decimal places when appropriate.

Deliverables

1. Choose a Topic where you can gather at least 50 pieces of data.

Examples of Topics

1. The Golden Gate Warriors Points Per Game in 2016 (use the points scored in the first 50 games).
2. High School Graduation Rates by State (use the graduation rates for all 50 states)
3. Average Tuition Rates in the US (You have to find the tuition rates of 50 college/universities).
4. The prices of a hotel room per night in a major city (You have to find the price of the same night of hotels in one city).
5. Weights of 50 babies at birth.

2. Write at least a 1-Page Report

Open a Word Document

1. Introduction--Provide a description of your topic and cite where you found your data.
2. Sample Data—Include a 5x10 table including your 50 values in your report. You must provide ALL of your sample data.
3. Problem Computations—For the topic you chose, you must answer the following:

• Determine the mean and standard deviation of your sample.
• Find the 80%, 95%, and 99% confidence intervals.
• Make sure to list the margin of error for the 80%, 95%, and 99% confidence interval.
• Create your own confidence interval (you cannot use 80%, 95%, and 99%) and make sure to show your work. Make sure to list the margin of error.

4. Problem Analysis—Write a half-page reflection.

• What trend do you see takes place to the confidence interval as the confidence level rises? Explain mathematically why that takes place.
• Provide a sentence for each confidence interval created in part c) which explains what the confidence interval means in context of topic of your project.
• Explain how Part I of the project has helped you understand confidence intervals better?
• How did this project help you understand statistics better?

Answer 1

Answer:

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%

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