For Each Of The 2 Majors, Consider The School Type Column

For Each Of The 2 Majors Consider The School Type Column Const

Construct a 90% confidence interval for the proportion of schools that are ‘Private’ for each of the two majors, considering the ‘School Type’ column. Additionally, construct a 99% confidence interval for the mean ‘Annual % ROI’ for each major. Provide interpretation and comparison of these intervals, and discuss their relevance to students choosing majors.

Paper For Above instruction

The purpose of this analysis is to determine the proportion of private schools and the average return on investment (ROI) for students in two different majors—business and engineering—using confidence intervals to estimate population parameters based on sample data. Understanding these intervals helps students make more informed decisions regarding their education and future earnings potential, taking into account the variability inherent in sample-based estimates.

Introduction

Statistical inference plays a vital role in education and career decision-making, especially when it comes to understanding factors like school type and potential financial returns. Confidence intervals provide a range of plausible values for population parameters, facilitating better risk assessment and decision-making. In this context, we analyze the proportion of private schools and the mean ROI for business and engineering majors, based on sample data, to give students a clearer picture of the landscape they are entering.

Analysis of School Type: Proportion of Private Schools

To evaluate the proportion of private schools within each major, we first extract the relevant sample data. Suppose, in Week 1, we found that among the sample of business schools, 65 out of 130 schools were private, resulting in a sample proportion of 0.5 (50%). For engineering schools, 70 out of 140 were private, again a sample proportion of 0.5.

Using this data, we construct a 90% confidence interval for each proportion. The formula for the confidence interval of a population proportion is:

CI = p̂ ± Z*(√[p̂(1 - p̂) / n])

where p̂ is the sample proportion, n is the sample size, and Z is the Z-score corresponding to the confidence level (for 90%, Z ≈ 1.645).

Calculations for business schools:

• p̂ = 0.5
• n = 130
• Standard error = √[0.5 * 0.5 / 130] ≈ 0.0439
• Margin of error = 1.645 * 0.0439 ≈ 0.0724
• Confidence interval: 0.5 ± 0.0724 → (0.4276, 0.5724)

Similarly, for engineering schools:

• p̂ = 0.5
• n = 140
• Standard error ≈ 0.0424
• Margin of error ≈ 1.645 * 0.0424 ≈ 0.0698
• Confidence interval: 0.5 ± 0.0698 → (0.4302, 0.5698)

These intervals suggest that between approximately 42.8% to 57.2% of schools are private for business, and between 43.0% to 57.0% for engineering, with 90% confidence that the actual proportions fall within these ranges.

Interpretation and Practical Implications

In Week 1, the percentage of private schools was directly observed—for example, 50% in both majors. However, this sample percentage alone does not account for sampling variability. The confidence interval provides a range where the true proportion likely resides, with a specified level of confidence (90% in this case).

The interval offers more nuanced information, reflecting potential sampling errors or uncertainties. For students, this means that if they are considering the likelihood of enrolling in a private school, they should recognize that the exact proportion could vary within this interval. The 90% confidence level indicates that if many samples were taken, approximately 90% of such intervals would contain the true population proportion.

This approach is more informative than a single sample percentage because it incorporates uncertainty, allowing students to evaluate risks and expectations more realistically when choosing a major or school type.

Comparison of Confidence Intervals and ROI Analysis

Next, to assess the financial outlook for students, we construct 99% confidence intervals for the mean ‘Annual % ROI’ for each major. Suppose we observe from the sample data that the average ROI for business majors is 8%, with a standard deviation of 2.5%, and for engineering majors, the average ROI is 10%, with a standard deviation of 3%. With sample sizes of 130 for business and 140 for engineering, the calculations proceed as follows.

For the mean ROI in business majors:

• Sample mean = 8%
• Standard deviation (s) = 2.5%
• Sample size (n) = 130
• Standard error = s / √n ≈ 2.5 / √130 ≈ 0.219
• Z-score for 99% confidence ≈ 2.576
• Margin of error = 2.576 * 0.219 ≈ 0.565
• Confidence interval: 8% ± 0.565% → (7.435%, 8.565%)

For engineering majors:

• Sample mean = 10%
• Standard deviation = 3%
• Standard error ≈ 3 / √140 ≈ 0.253
• Margin of error ≈ 2.576 * 0.253 ≈ 0.652%
• Confidence interval: 10% ± 0.652% → (9.348%, 10.652%)

The resulting intervals suggest that the true average ROI for business majors likely falls between approximately 7.4% and 8.6%, whereas for engineering it is between 9.3% and 10.7%, with 99% confidence.

Interpretation of Confidence Intervals and Decision-Making

These intervals are critical because they visually represent the range within which the true population mean ROI is likely to be, incorporating uncertainty due to sampling variability. If we knew the exact mean ROI, a confidence interval would still serve as an estimate with a specified confidence level, but it does not guarantee that the true mean lies within it. Instead, it indicates that, over many repeated samples, approximately 99% of the calculated intervals would contain the true mean.

Comparing the intervals for the two majors, the engineering ROI appears higher and provides a wider range. Such information helps students decide which major might offer better financial returns. The engineering ROI interval gives more confidence that the ROI will be above a certain threshold, aiding students concerned about maximizing their investment.

In practical terms, the narrower and higher interval for engineering suggests it could be a better investment option in terms of ROI, although other factors such as individual interests, job market conditions, and personal skills should also weigh into decision-making.

Conclusion

Constructing confidence intervals for both proportions of private schools and the mean ROI provides valuable insights into the variability and reliability of sample estimates. These intervals help students anticipate the range of likely real-world outcomes and make more informed educational and financial decisions. While the proportion intervals guide expectations regarding school types, the ROI intervals focus on potential earning power after graduation. Overall, confidence intervals are essential tools that enhance understanding beyond point estimates, fostering better preparedness and strategic planning for students navigating the choice of majors and institutions.

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