3.11 Question bank

No answers are given for this section, by design. These are for testing yourself once the practice problems are complete.

Multiple choice

QB 3.1 The divisor \(n - 1\) in the sample variance is used because

  1. it makes the variance larger, which is safer
  2. \(\bar{x}\) was estimated from the same data, leaving \(n-1\) free deviations
  3. the population is finite
  4. the sample standard deviation would otherwise be negative

QB 3.2 A 95% confidence interval for \(\mu\) is \([112, 128]\). Which statement is correct?

  1. 95% of the population lies between 112 and 128
  2. There is a 95% probability that \(\mu\) lies between 112 and 128
  3. 95% of intervals built this way would contain \(\mu\)
  4. The sample mean has a 95% chance of lying between 112 and 128

QB 3.3 Holding everything else fixed, moving from 95% to 99% confidence

  1. narrows the interval
  2. widens the interval
  3. leaves the width unchanged but shifts the centre
  4. requires a larger sample

QB 3.4 For a sample proportion, the standard error is largest when \(\hat{p}\) is closest to

  1. 0
  2. 0.25
  3. 0.5
  4. 1

QB 3.5 A survey of 600 households finds 30% own a two-wheeler. The approximate 95% margin of error is

  1. 0.4 percentage points
  2. 1.9 percentage points
  3. 3.7 percentage points
  4. 7.5 percentage points

QB 3.6 The \(t\) distribution is used instead of the normal when

  1. the sample is small
  2. \(\sigma\) is estimated from the sample
  3. the population is skewed
  4. the confidence level exceeds 95%

QB 3.7 To halve the margin of error of a survey, the sample size must be multiplied by

  1. \(\sqrt{2}\)
  2. 2
  3. 4
  4. It depends on the confidence level

QB 3.8 Which of the following would not narrow a confidence interval for \(\mu\)? (select all that apply)

  1. Increasing the sample size
  2. Lowering the confidence level
  3. Sampling from a population with smaller \(\sigma\)
  4. Sampling only from a subgroup that happens to be homogeneous

Short reasoning

QB 3.9 A report gives a 95% confidence interval for mean household income of [₹18,200, ₹19,600] and concludes: “almost all households in this district earn between ₹18,200 and ₹19,600.” Identify the error and state what the interval does say.

QB 3.10 Explain why the population size does not appear in the formula for the margin of error, and why this surprises people.

QB 3.11 Two districts are surveyed. District A’s 95% interval for mean income is [₹19,000, ₹23,000]; District B’s is [₹22,000, ₹26,000]. A journalist writes that the districts do not differ significantly because the intervals overlap. Evaluate.

QB 3.12 A researcher has a sample of 12 observations from a strongly skewed population and constructs a \(t\) interval. State two distinct assumptions being relied on, and say which one the \(t\) distribution addresses.

QB 3.13 Explain why \(s\) is not an unbiased estimator of \(\sigma\) even though \(s^2\) is unbiased for \(\sigma^2\), and why this is rarely mentioned in practice.

QB 3.14 A polling organisation reports a margin of error of ±3 points but sampled only households with landline telephones. Explain what the ±3 does and does not cover.

Numerical

QB 3.15 A sample of 16 bags of rice has mean weight 24.6 kg and sample standard deviation 1.2 kg.

  1. Construct a 95% confidence interval for the mean weight.
  2. Construct a 99% confidence interval.
  3. The label claims 25 kg. Comment.
  4. How would your answer to (i) change if the standard deviation of 1.2 kg were known to be the population value?

QB 3.16 In a random sample of 250 farm households in a district, 145 reported using irrigation.

  1. Estimate the proportion and its standard error.
  2. Construct a 90% confidence interval.
  3. Construct a 95% lower confidence bound.
  4. A scheme guarantees that at least half of households have irrigation. Is the sample consistent with that?

QB 3.17 A state government wishes to estimate mean annual household expenditure to within ₹500 at 95% confidence. Previous rounds suggest \(\sigma \approx ₹9{,}000\).

  1. What sample size is required?
  2. Fieldwork costs ₹1,100 per household. What is the budget?
  3. The budget is capped at ₹15 lakh. What margin of error is achievable?

QB 3.18 Using data/wages-india-synthetic.csv, write R code to construct 95% confidence intervals for mean annual earnings separately for urban and rural workers, using samples of 300 from each. Report both intervals, state whether they overlap, and explain carefully what that does and does not tell you about the difference between the two groups.