4.13 Question bank

No solutions are given for this section, by design. These questions are drawn from past quizzes and exams. If you can work them without checking an answer key, you are ready; if you cannot, the section to reread is named beside each one.

They run from easiest to hardest. Do them in order.

Multiple choice

QB 11.1 The level of significance, \(\alpha\), is the risk of (select all that apply)

  1. Rejecting the null hypothesis when it is true
  2. Rejecting the null hypothesis when the alternative hypothesis is true
  3. Rejecting the alternative hypothesis when the null hypothesis is true
  4. Rejecting the alternative hypothesis when it is true

See Section 4.4.

QB 11.2 An economist sets the alternative hypothesis that the average CGPA of undergraduate students is less than 8. What type of test should be used, and what is the critical value at \(\alpha = 5\%\)?

  1. One-tailed test and \(+1.96\)
  2. One-tailed test and \(-1.96\)
  3. One-tailed test and \(+1.645\)
  4. One-tailed test and \(-1.645\)

See Section 4.8.

QB 11.3 The margin of error for a confidence interval is \(\pm Z_{\alpha/2}\,\sigma/\sqrt{n}\). Which statements are correct? (select all that apply)

  1. Increasing the sample size will decrease the margin of error
  2. The margin of error decreases as \(\alpha\) rises
  3. The larger the margin of error, the less confident one is about the estimate
  4. The margin of error increases with the population standard deviation \(\sigma\)

See Section 4.6.

QB 11.4 A 95% confidence interval for mean student height is \((165.41,\ 170.95)\). You now compute a 99% interval from the same data. Which statements are likely correct? (select all that apply)

  1. The 90%, 95% and 99% intervals are the same
  2. The 99% interval is \((164.54,\ 171.83)\)
  3. The 99% interval is \((165.86,\ 170.51)\)
  4. The answer cannot be determined from the information given

See Section 4.6. Ask yourself which way an interval must move when you demand more confidence.

QB 11.5 A company tests the null hypothesis that the average time to accept a job offer is at most 7 days. The test concludes that the average exceeds 7 days. It is later learned that the true mean is exactly 7 days. What error occurred?

  1. Type II error
  2. Type I error
  3. A correct decision was made
  4. Cannot be determined from the information provided

Short reasoning

QB 11.6 “A high \(p\)-value is evidence that the null hypothesis is true.” Do you agree? Why or why not?

See Section 4.3.

QB 11.7 Two economists, Amit and Anand, collect data on Indian agricultural wages to test the effect of MNREGA. Amit concludes that wages after the policy are higher than before, when in fact wages have not changed. Anand concludes that wages before and after are the same — also incorrectly.

It is claimed that Amit has committed a Type I error and Anand a Type II error. Do you agree? Why or why not?

Numerical

QB 11.8 A researcher wants to estimate the average weekly study hours of university students. The population standard deviation is 6 hours. What sample size is needed to estimate the mean with a margin of error of 1 hour at 90% confidence?

  1. \(n = 98\)
  2. \(n = 64\)
  3. \(n = 121\)
  4. \(n = 82\)

QB 11.9 The numbers of people taking the morning university shuttle on 16 randomly chosen days are:

27, 46, 35, 33, 29, 45, 28, 24, 30, 40, 38, 35, 41, 27, 38, 29

The administration claims the daily average is 30. Taking \(\sigma = 7\):

  1. Construct a 99% confidence interval for the mean number of riders.
  2. Use that interval to test the administration’s claim.
  3. Would your conclusion change at the 95% level? Explain why before you compute it.

Part (c) is the one that separates understanding from arithmetic.

Errors and power

QB 4.18 A Type II error is

  1. rejecting a true null hypothesis
  2. failing to reject a false null hypothesis
  3. choosing the wrong significance level
  4. computing the \(p\)-value incorrectly

QB 4.19 Reducing \(\alpha\) from 0.05 to 0.01, holding everything else fixed,

  1. reduces both error probabilities
  2. reduces Type I error and increases Type II error
  3. increases Type I error and reduces Type II error
  4. leaves power unchanged

QB 4.20 The power of a test is

  1. \(\alpha\)
  2. \(1 - \alpha\)
  3. \(\beta\)
  4. \(1 - \beta\)

QB 4.21 Power increases when (select all that apply)

  1. the sample size grows
  2. the true effect is larger
  3. the population standard deviation is larger
  4. \(\alpha\) is made smaller

QB 4.22 A study reports “no significant effect” and had 30% power against the effect it was looking for. The most accurate conclusion is

  1. there is no effect
  2. the effect is small
  3. the study was unlikely to detect the effect even if it existed
  4. the null hypothesis has been confirmed

QB 4.23 In the Decision Review System, “umpire’s call” on a marginal ball-tracking decision reflects a choice to

  1. keep \(\alpha\) small
  2. keep \(\beta\) small
  3. maximise power
  4. eliminate both errors

QB 4.24 Explain why power cannot be stated for a test without also stating an effect size, whereas \(\alpha\) can.

QB 4.25 A colleague computes the power of a study after it has returned a null result, using the effect size the study itself estimated. Explain what is wrong with this.

QB 4.26 A quality inspector tests whether a machine’s mean fill weight has drifted from 500 g. With \(\sigma = 8\) g, \(n = 25\) and \(\alpha = 0.05\) two-sided:

  1. State the critical values.
  2. Compute the power against a true mean of 503 g.
  3. Compute the power against a true mean of 505 g.
  4. How large a sample would give 90% power against a 3 g drift?

QB 4.27 Two studies test the same hypothesis. Study A has 40 observations, Study B has 400. Both report \(p = 0.04\).

  1. Which provides stronger evidence that the effect is large?
  2. Which had more power against a given effect size?
  3. Explain why these two questions have different answers.