Daily 6 - Feb 4

Class Performance

Students: 109 | Mean: 4.40 | Median: 4.5 | SD: 0.66

Scores ranged from 2.5 to 5 out of 5 points.

Score Distribution

Performance by Question

Questions

Q1: Estimating variance — plug in _____ for E(X) and replace outer expectation with sample _____

\(\overline{X}\) (sample average); average — Plug in the sample mean for \(E(X)\) and replace the outer expectation with another sample average.

  • Near-universal success — Almost every student correctly identified \(\overline{X}\) and “average.”
  • Minor notation variants — Some wrote “x-bar” or “mean(x)” instead of \(\overline{X}\); all accepted as equivalent.
  • No significant errors observed — This was the strongest question across the class.

Q2: True or false — if an estimator is unbiased, then it will equal the estimand

False. An unbiased estimator equals the estimand on average across repeated samples, not necessarily for any single estimate.

  • Answering “True” — The most common error. Students confused “unbiased on average” with “always equals the true value.”
  • Crossed-out corrections — Several students initially wrote one answer then changed, suggesting uncertainty about this concept.
  • Key distinction — Any single estimate can differ from the estimand due to sampling variability; unbiasedness is a property of the sampling distribution.

Q3: Explain what summarise does in the “Estimating the CEF” code chunk

It computes the sample average of earnings for each age group — estimating the conditional expectation function (CEF).

  • Missing “by age” — Many mentioned “mean earnings” or “average” but omitted that it groups by age. The grouping is central to estimating the CEF.
  • Too vague — Answers like “simplifies the data,” “creates a table,” “displays data better,” or “compiles data” don’t capture the computational step.
  • Describing the concept, not the computation — Some wrote about “conditional expectations” abstractly without stating the specific operation: computing mean earnings grouped by age.

Q4: The estimated CEF in Figure 6 has a _____ shape

Concave — The estimated CEF shows a concave (inverted U) relationship between age and earnings.

  • Near-universal success — Almost every student correctly identified the concave shape.
  • Rare error: “variance” — One student wrote “variance” instead of describing the shape.
  • No other significant errors — This was one of the strongest questions alongside Q1.

Q5: That shape is well-fit by a _____ in age

Quadratic — The concave shape is well-fit by a quadratic (second-degree polynomial) in age.

  • Answering “polynomial” — While quadratic is a polynomial, the specific answer is quadratic. “Polynomial” is too broad.
  • Answering “increase” or “condition” — These don’t describe a functional form.
  • Overall strong performance — The vast majority correctly identified “quadratic.”

Key Takeaways

Strengths: Variance estimation plug-ins (Q1), concave CEF shape (Q4), quadratic fit (Q5).

Review:

  • Unbiased ≠ always correct — unbiased means correct on average across samples, not for every single estimate
  • summarise computes average earnings by age — not just “displays” or “organizes” data; it calculates the mean within each group
  • Be specific about functional forms — “quadratic” is more precise than “polynomial”