1.11 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 1.1 Which of the following is a random variable? (select all that apply)

  1. The number showing after a die is rolled
  2. The number showing on a die with a three painted on every face
  3. The mean earnings of a sample of 500 workers, before the sample is drawn
  4. The mean earnings of a sample of 500 workers, after the sample is drawn

QB 1.2 A population has mean \(\mu = 40\) and standard deviation \(\sigma = 12\). For samples of size 36, the standard deviation of the sample mean is

  1. 12
  2. 2
  3. 0.33
  4. 6

QB 1.3 Which statements about the Law of Large Numbers are correct? (select all that apply)

  1. As the sample grows, the sample mean becomes more likely to be close to the population mean
  2. As the sample grows, the sample mean becomes certain to equal the population mean
  3. It applies only when the population is normally distributed
  4. It explains why a survey need not interview everyone

QB 1.4 A researcher doubles the size of a survey from 500 to 1,000 households. The standard deviation of the sample mean falls by a factor of approximately

  1. 2
  2. 4
  3. 1.41
  4. It does not change

QB 1.5 The Central Limit Theorem states that

  1. Large samples are normally distributed
  2. The population becomes normal as the sample grows
  3. The sampling distribution of the sample mean is approximately normal for sufficiently large samples
  4. Any variable is approximately normal if enough observations are collected

QB 1.6 Which of the following would make the normal approximation to the sampling distribution less accurate at a given sample size? (select all that apply)

  1. A heavily right-skewed population
  2. A symmetric population
  3. A population with a small number of extreme outliers
  4. A larger population standard deviation

QB 1.7 A statistic differs from a parameter because

  1. A statistic is always correct
  2. A statistic is computed from a sample and changes from sample to sample
  3. A parameter is computed from a sample
  4. A parameter changes when a different sample is drawn

Short reasoning

QB 1.8 A newspaper reports that average household income in a district is ₹32,000, based on a survey of 800 households, and describes this as “the income of a typical household”. Give two separate objections to that description.

QB 1.9 An analyst has a sample of 50,000 observations and concludes that because the sample is very large, the results must be reliable. Under what circumstances would this reasoning be wrong?

QB 1.10 The gambler’s fallacy is the mistaken belief that chance events “balance out” in the short run.

A batsman has scored 0 runs in each of his last five innings. A commentator says:

He is due for a big score today.

Explain why this may be an example of the gambler’s fallacy, and what the Law of Large Numbers does and does not say about it. Under what circumstances could the commentator’s statement nevertheless be reasonable?

QB 1.11 Explain why the sampling distribution of the sample mean is narrower than the distribution of the individual observations, using the idea of offsetting variation rather than any formula.

Numerical

QB 1.12 The weights of rice bags filled by a machine have mean 25 kg and standard deviation 0.6 kg. A quality inspector weighs 36 bags.

  1. What is the expected value of the sample mean weight?
  2. What is its standard deviation?
  3. How many bags would need to be weighed to halve that standard deviation?

QB 1.13 Daily wages of agricultural labourers in a district have mean ₹350 and standard deviation ₹90. For a random sample of 100 labourers, approximate the probability that the sample mean wage

  1. exceeds ₹360, (b) falls below ₹340, (c) lies between ₹340 and ₹360.

QB 1.14 A population is strongly right-skewed with mean 12 and standard deviation 15. A student proposes to use the normal approximation for the sampling distribution of the mean with a sample of 20 observations. Comment on whether this is advisable, and describe how you would check.