2.10 Question bank

No answers are given for this section, by design. These questions are for testing yourself once the practice problems are complete. There are no hints and no worked solutions — exactly as in an examination.

Multiple choice

QB 2.1 Which of the following is an estimate rather than an estimator?

  1. \(\bar{X}\)
  2. The rule “take the median of the sample”
  3. ₹18,400
  4. \(\sigma/\sqrt{n}\)

QB 2.2 An estimator is unbiased if

  1. every estimate it produces equals the parameter
  2. its expected value equals the parameter
  3. its variance falls as the sample size grows
  4. the sample is drawn at random

QB 2.3 A population has \(\sigma = 60\). For the standard error of the sample mean to equal 2, the sample size must be

  1. 30
  2. 120
  3. 900
  4. 3,600

QB 2.4 A survey samples only households with telephones. Which is the most accurate description of the consequence?

  1. The estimates will be centred correctly but too variable
  2. The estimates will be centred on the wrong value, and a larger sample will not fix it
  3. The estimates will be centred on the wrong value, but a larger sample will fix it
  4. Nothing goes wrong provided the sample is large

QB 2.5 The variance derivation \(\mathrm{Var}(\bar{X}) = \sigma^2/n\) uses independence at exactly one step. Which one?

  1. Taking the constant \(1/n\) outside the variance
  2. Writing \(\mathrm{Var}(aX) = a^2 \mathrm{Var}(X)\)
  3. Replacing the variance of the sum with the sum of the variances
  4. Cancelling \(n\) against \(n^2\)

QB 2.6 Two surveys estimate the same quantity. Survey X has a standard error of ₹4,000; Survey Y has a standard error of ₹1,000. Which statement follows? (select all that apply)

  1. Survey Y’s estimate is closer to the truth
  2. Survey Y’s estimates vary less across repeated sampling
  3. Survey Y used a larger sample, if both were drawn the same way
  4. Survey Y is unbiased

QB 2.7 Doubling a survey’s sample size reduces its standard error by a factor of

  1. 4
  2. 2
  3. \(\sqrt{2}\)
  4. It depends on \(\sigma\)

Short reasoning

QB 2.8 A student writes: “the standard error tells us how much our estimate differs from the population mean.” Correct the statement, and explain in one sentence why the original cannot be right.

QB 2.9 Explain why an estimator with zero variance is not necessarily a good estimator. Give an example.

QB 2.10 A researcher collects 500 households by standing outside a government hospital on a weekday morning and interviewing whoever passes. Identify at least two distinct problems with this sample, and state for each whether it affects the centre of the estimates, their spread, or both.

QB 2.11 Two rules for estimating \(\mu\) from \(n\) observations are proposed: the sample mean, and the average of the first \(n/2\) observations. Both are unbiased. Explain how you would decide between them, and what the answer is.

QB 2.12 An economist reports that increasing a survey from 1,000 to 1,200 households “substantially improved the precision of our estimates.” Evaluate that claim quantitatively.

QB 2.13 Section 2.2 showed a badly designed survey producing estimates with a smaller spread than a well-designed one. Explain how this is possible, and what it implies about using a reported standard error to judge whether a study is trustworthy.

Numerical

QB 2.14 A population of farm households has mean annual income \(\mu = ₹84{,}000\) and standard deviation \(\sigma = ₹36{,}000\).

  1. Find the standard error of the sample mean for a survey of 144 households.
  2. What sample size would be needed to reduce the standard error to ₹1,500?
  3. Fieldwork costs ₹1,200 per household. What is the additional cost of the sample in (ii) relative to the one in (i)?

QB 2.15 For a population with variance \(\sigma^2\), consider the estimator \(W = \tfrac{1}{3}X_1 + \tfrac{2}{3}X_2\) computed from two independent observations.

  1. Show that \(W\) is unbiased.
  2. Find \(\mathrm{Var}(W)\).
  3. Compare it with the variance of the ordinary sample mean of the same two observations. Which is preferable, and what general principle does this illustrate?

QB 2.16 A state has 40 districts. A survey selects 4 districts at random and interviews 250 households in each, reporting a standard error of \(\sigma/\sqrt{1000}\).

  1. State the assumption this calculation requires.
  2. Explain why it is unlikely to hold here.
  3. Say whether the reported standard error is likely to be too large or too small, and why that direction is the dangerous one.

QB 2.17 Using data/wages-india-synthetic.csv as a population, write R code to estimate how large a sample is needed before the standard error of mean annual earnings falls below ₹4,000. Verify your answer by simulation as well as by formula, and explain any discrepancy between the two.