Using The ROI Data Set Excel Document – Please Explain The S
Using The Roi Data Set Excel Doc Please Explain The Steps1 For Each
Using the ROI data set: Excel doc-Please explain the steps 1. For each of the 2 majors consider the ‘School Type’ column. Assuming the requirements are met, construct a 90% confidence interval for the proportion of the schools that are ‘Private’. Be sure to interpret your results. 2. For each of the 2 majors construct a 95% confidence interval for the mean of the column ‘Annual % ROI’. Be sure to interpret your results.
Paper For Above instruction
Introduction
The data derived from the ROI dataset provides valuable insights into the distribution and performance of schools based on their type and major. The primary objectives are to determine the proportion of private schools within each major and to estimate the average annual return on investment (ROI) for each major through confidence intervals. These statistical procedures enable stakeholders to make informed decisions regarding educational investments and understand the variability in ROI across different school types and majors.
Constructing a 90% Confidence Interval for the Proportion of Private Schools for Each Major
The first task involves analyzing the ‘School Type’ column for each major, specifically to estimate the proportion of schools classified as ‘Private’. The steps are as follows:
1. Data Segmentation: Filter the dataset to separate the entries based on majors (Business and Engineering). Within each major, further categorize the schools based on ‘School Type’ (Private or Public).
2. Calculating Sample Proportions: For each major, count the number of private schools (\(x\)) and the total number of schools (\(n\)) considered. The sample proportion (\(\hat{p}\)) is computed as:
\[
\hat{p} = \frac{x}{n}
\]
3. Verify Conditions: Ensure the sample size is sufficiently large for the normal approximation to be valid. The common rule is that both \(x\) and \(n - x\) should be at least 10.
4. Determine the Z-value for 90% Confidence: The Z-score corresponding to a 90% confidence level (two-tailed) is approximately 1.645.
5. Compute Margin of Error (ME): The margin of error for the proportion is given by:
\[
ME = Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
6. Calculate Confidence Interval: The interval is:
\[
\left(\hat{p} - ME, \hat{p} + ME \right)
\]
7. Interpretation: Conclude that, with 90% confidence, the true proportion of private schools in each major lies within this interval, providing insight into school type distribution within each major.
For example, if within the Business major, 10 schools out of 20 are private, then:
\[
\hat{p} = \frac{10}{20} = 0.5
\]
The margin of error is:
\[
ME = 1.645 \times \sqrt{\frac{0.5 \times 0.5}{20}} \approx 1.645 \times 0.1118 \approx 0.184
\]
Thus, the 90% CI is approximately (0.316, 0.684), indicating that there's a 90% probability the true proportion of private Business schools is between 31.6% and 68.4%.
Constructing a 95% Confidence Interval for the Mean ‘Annual % ROI’ for Each Major
The second task involves estimating the average ‘Annual % ROI’ for each major with a 95% confidence interval. The steps are:
1. Data Extraction: For each major, compile all ‘Annual % ROI’ values of the respective schools.
2. Calculate Sample Mean (\(\bar{x}\)) and Sample Standard Deviation (s):
\[
\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
\]
\[
s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
\]
3. Confirm Normality or Use Large Sample Properties: With sufficiently large samples, the Central Limit Theorem justifies the use of the normal approximation.
4. Determine the t-value for 95% Confidence: Using degrees of freedom \(df = n-1\), find the critical t-value from the t-distribution table.
5. Calculate Margin of Error (ME):
\[
ME = t \times \frac{s}{\sqrt{n}}
\]
6. Calculate Confidence Interval:
\[
(\bar{x} - ME, \bar{x} + ME)
\]
7. Interpretation: The interval estimates the range within which the true mean of ‘Annual % ROI’ in each major likely falls with 95% confidence. For example, a mean of 7.5% with a margin of 0.5% indicates the true mean ROI is probably between 7.0% and 8.0%.
In application, practice should involve performing these calculations with actual data values from the dataset, which can be easily done using Excel functions such as AVERAGE(), STDEV.S(), and T.INV.2T() for t-values.
Conclusion
The statistical techniques of confidence interval estimation are vital for interpreting the proportions and means in educational ROI data. Constructing these intervals provides quantitative measures of uncertainty, essential for informed decision-making. For the ROI dataset, the proportion of private schools within majors and the average annual ROI are key indicators of the educational investment landscape. Accurate calculations and interpretations of these intervals enable stakeholders to assess the prevalence of private education options and their financial outcomes with specified confidence levels, fostering better planning and evaluation.
References
- Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
- Agresti, A. (2018). An Introduction to Categorical Data Analysis. Wiley.
- Devore, J. L., & Peck, R. (2017). Statistics: The Exploration and Analysis of Data. Cengage Learning.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
- Johnson, R. A., & Wichern, D. W. (2019). Applied Multivariate Statistical Analysis. Pearson.
- Larsen, R. J., & Marx, M. L. (2012). An Introduction to Mathematical Statistics and Its Applications. Pearson.
- Student t-distribution: https://statisticsbyjim.com/hypothesis-testing/t-distributions-and-confidence-intervals/
- Excel functions for statistical analysis: https://support.microsoft.com/en-us/excel
- Payscale. (2013). Best College ROI by Major. https://www.payscale.com/college-roi
- U.S. Census Bureau. (2020). Statistical Data and Analysis. https://www.census.gov/data.html