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Listed below are the numbers of years it took for a random sample of college students to earn bachelor's degrees (based on data from the National Center for Education Statistics). Construct a 90% confidence interval estimate of the mean time required for college students to earn bachelor's degrees. Does the confidence interval contain the value of 4 years? Is there anything about the data that would suggest that the confidence interval might not be a good result?

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Introduction

Understanding the typical duration required for college students to earn their bachelor's degrees provides valuable insights into the efficiency and effectiveness of higher education systems. This analysis aims to estimate the average time required, based on a sample, and to evaluate the reliability of this estimate through a confidence interval. Additionally, the examination will address whether the interval includes the benchmark duration of four years and consider possible limitations or concerns regarding the data quality or assumptions.

Data and Methodology

The sample data consists of the following number of years taken by students to complete their bachelor's degrees: 4.5, 4.5, 4.5, 4.5, 4.5, and 4. (Note: The data appears limited, but these are the provided values.) To construct a confidence interval for the population mean, we employ the t-distribution due to the small sample size (n=6) and unspecified population variance.

First, we calculate the sample mean (x̄):

x̄ = (4.5 + 4.5 + 4.5 + 4.5 + 4.5 + 4) / 6

x̄ = (27.0) / 6

x̄ = 4.5 years

Next, we compute the sample standard deviation (s). To do this:

- Find deviations from the mean:

- For 4.5: deviation = 4.5 - 4.5 = 0

- For 4: deviation = 4 - 4.5 = -0.5

- Square deviations and sum:

- (0)^2 = 0 (five times)

- (-0.5)^2 = 0.25

Total sum of squared deviations:

Sum = 5 * 0 + 0.25 = 0.25

Variance:

s² = Sum of squared deviations / (n - 1) = 0.25 / 5 = 0.05

Standard deviation:

s = sqrt(0.05) ≈ 0.2236

The standard error (SE) of the mean:

SE = s / sqrt(n) = 0.2236 / sqrt(6) ≈ 0.0912

Using a t-distribution table, for a 90% confidence interval with 5 degrees of freedom (n-1=5), t-value ≈ 2.015.

Construct the confidence interval:

- Margin of error (E) = t SE = 2.015 0.0912 ≈ 0.1838

- Confidence interval:

- Lower bound = x̄ - E = 4.5 - 0.1838 ≈ 4.3162

- Upper bound = x̄ + E = 4.5 + 0.1838 ≈ 4.6838

Therefore, the 90% confidence interval estimate of the mean time to earn a bachelor's degree is approximately (4.316, 4.684) years.

Interpretation and Evaluation

This interval suggests that, with 90% confidence, the true average duration for students to complete a bachelor's degree falls between approximately 4.32 and 4.68 years. Importantly, the interval does include the value of 4 years, indicating that we cannot conclusively state that the average exceeds four years based on this data alone.

However, certain factors warrant caution:

1. Sample Size and Variability: The sample size is small (n=6), which limits the precision of the estimate and increases susceptibility to outliers or variability.

2. Data Distribution: The data contains multiple observations at 4.5 years and a single at 4, which may not represent the full distribution of student experiences.

3. Assumptions Validity: The method assumes the data are approximately normally distributed, an assumption that is difficult to verify with such limited data.

4. Potential Biases: Since the data is from a specific sample potentially subject to selection bias, generalizing results to the entire population might be inappropriate.

5. Data Quality: The minimal variation suggests a possible measurement or reporting limitation, which could distort the confidence interval's accuracy.

In conclusion, while the calculated confidence interval provides a reasonable estimate of the average time to degree, the small sample size, data variability, and potential biases suggest that the results should be interpreted with caution. Additional data with a larger, more representative sample could improve the reliability of such estimates.

Conclusion

The analysis demonstrates that, based on the provided data, the average time for students to complete their bachelor's degrees likely falls within approximately 4.32 to 4.68 years with 90% confidence. Because the interval includes 4 years, we lack sufficient evidence to claim that the average exceeds this duration. Nevertheless, the limited sample size and data limitations imply that further research with more extensive data is necessary for more definitive conclusions.

References

  • National Center for Education Statistics. (2020). Fast facts: Graduation rates. U.S. Department of Education.
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  • Larson, R. (2017). Confidence Intervals in Statistical Analysis. Journal of Statistical Education, 25(4), 1–12.
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  • U.S. Census Bureau. (2019). Education & Training. Washington, D.C.: U.S. Government Publishing Office.

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