What is a standard error and why is it important? [Includes sub-question with failure data]

Part a) What is Standard Error and Why is it Important?

Definition:

The Standard Error (SE) is a statistical term that measures the accuracy with which a sample distribution represents a population. In simpler terms, it tells us how much the sample mean (or proportion) is likely to vary from the actual population mean (or proportion).

Formula:

For sample mean: SE = σ / √n

Where:

• σ = population standard deviation
• n = sample size

For sample proportion: SE = √[p(1−p)/n]

Importance of Standard Error:

• 1. Measures Sampling Accuracy: It shows how accurately the sample estimates the population parameter. Smaller SE means the sample is a good representation.
• 2. Confidence Intervals: SE is used to construct confidence intervals, which show the range where we expect the true population value to lie.
• 3. Hypothesis Testing: SE is used to calculate test statistics like Z-scores or t-scores to determine the significance of results.
• 4. Comparison of Groups: It helps in comparing differences between sample means or proportions from different groups.
• 5. Understanding Variability: A high SE indicates high variability or less reliability in the sample estimate.

Example:

If the average exam score in a sample of students is 65 with an SE of 2, we can say the true average score is likely between 63 and 67 (for 95% confidence).

Part b) Compute Standard Error of Difference Between Proportions

Given:

• University Teaching Departments: n₁ = 400, failures = 300 → p₁ = 300/400 = 0.75
• Affiliated Colleges: n₂ = 500, failures = 300 → p₂ = 300/500 = 0.60

Standard Error of Difference Between Proportions:

Formula:

SE = √[(p₁(1 − p₁)/n₁) + (p₂(1 − p₂)/n₂)]

Step-by-Step Calculation:

• p₁ = 0.75, 1 − p₁ = 0.25, p₁(1 − p₁) = 0.75 × 0.25 = 0.1875
• p₂ = 0.60, 1 − p₂ = 0.40, p₂(1 − p₂) = 0.60 × 0.40 = 0.24
• 0.1875 / 400 = 0.00046875
• 0.24 / 500 = 0.00048
• SE = √(0.00046875 + 0.00048) = √0.00094875
• SE ≈ 0.0308

Interpretation:

The standard error of the difference between proportions of failures is approximately 0.0308 (or 3.08%).

This value can be used to construct a confidence interval or perform hypothesis testing to determine if the difference between the two proportions is statistically significant.

Conclusion

The Standard Error is an essential concept in statistics, helping researchers understand the reliability of sample estimates. It plays a critical role in hypothesis testing and confidence interval estimation. In the context of this problem, the SE of 0.0308 helps us measure the reliability of the observed difference in failure rates between university teaching departments and affiliated colleges.