5.9 Question bank

No answers are given for this section, by design.

Multiple choice

QB 5.1 The standard error of \(\bar{X} - \bar{Y}\) for independent samples is

  1. \(\dfrac{\sigma_x}{\sqrt{n}} - \dfrac{\sigma_y}{\sqrt{m}}\)
  2. \(\sqrt{\dfrac{\sigma_x^2}{n} + \dfrac{\sigma_y^2}{m}}\)
  3. \(\sqrt{\dfrac{\sigma_x^2}{n} - \dfrac{\sigma_y^2}{m}}\)
  4. \(\dfrac{\sigma_x + \sigma_y}{\sqrt{n + m}}\)

QB 5.2 The pooled two-sample \(t\)-test uses how many degrees of freedom?

  1. \(n + m\)
  2. \(n + m - 1\)
  3. \(n + m - 2\)
  4. \(\min(n, m) - 1\)

QB 5.3 A 95% confidence interval for \(\mu_x - \mu_y\) is \([-1.2,\ 4.8]\). At the 5% level we

  1. reject \(H_0: \mu_x = \mu_y\)
  2. fail to reject \(H_0: \mu_x = \mu_y\)
  3. cannot tell without the \(p\)-value
  4. conclude the two means are equal

QB 5.4 t.test(x, y) in R by default

  1. assumes equal variances
  2. performs Welch’s test
  3. performs a paired test
  4. performs a one-tailed test

QB 5.5 Which of the following are paired designs? (select all that apply)

  1. Blood pressure of 40 patients before and after a drug
  2. Test scores of 40 boys and 45 girls
  3. Yields on 20 plots under one seed and 20 different plots under another
  4. Household expenditure in the same 200 households in 2019 and 2023

QB 5.6 Ignoring pairing in a genuinely paired design usually

  1. makes it easier to reject the null
  2. makes it harder to reject the null
  3. has no effect on the \(p\)-value
  4. reverses the sign of the difference

QB 5.7 Two groups differ with \(p < 0.001\) in a sample of 50,000. This implies

  1. the difference is large
  2. the difference is important
  3. the difference is unlikely to be sampling variation alone
  4. one group caused the other’s outcome

Short reasoning

QB 5.8 Explain why subtracting one sample mean from another increases variability rather than reducing it, and give the algebraic reason.

QB 5.9 A researcher has two samples with \(s_x = 2\) and \(s_y = 9\), and uses the pooled test. State the assumption being made, why it is doubtful here, and what the consequence is.

QB 5.10 A newspaper reports that graduates of a coaching institute score 40 marks higher on average than non-attendees, with a very small \(p\)-value. Explain what the comparison can and cannot support.

QB 5.11 In a paired design, the two columns are strongly positively correlated. Explain, without algebra, why this makes the paired test more powerful than the unpaired one.

QB 5.12 A trial assigns a new fertiliser to plots by lottery. Explain why the difference in mean yields can be read as an effect here, when the same comparison in Section 5.6 could not.

QB 5.13 Two studies estimate the same difference. One reports \([0.1,\ 9.9]\) and the other \([4.8,\ 5.2]\). Both exclude zero. Explain what distinguishes them and which is more useful.

Numerical

QB 5.14 Government school students score a mean of 62.4 with \(s = 11.2\) in a sample of 45. Private school students score 66.1 with \(s = 9.8\) in a sample of 52.

  1. Compute the standard error of the difference.
  2. Compute the test statistic.
  3. Test at the 5% level whether the means differ.
  4. Construct a 95% confidence interval, and state whether it agrees with (iii).

QB 5.15 A sample of 8 plots under a new seed yields a mean of 41.2 quintals with \(s = 3.1\). A sample of 10 plots under the old seed yields 37.8 with \(s = 2.9\).

  1. Compute the pooled variance estimate and \(s_p\).
  2. Compute the pooled test statistic and its degrees of freedom.
  3. Test at the 5% level whether the new seed yields more, stating your hypotheses carefully.
  4. Would Welch’s test use more or fewer degrees of freedom here, and why?

QB 5.16 Twelve workers are timed on a task before and after training. The mean reduction is 4.2 minutes with a standard deviation of differences of 5.1 minutes.

  1. State the hypotheses for a test that training reduced completion time.
  2. Compute the test statistic and its degrees of freedom.
  3. Test at the 5% level.
  4. Explain why the standard deviation of the differences is the right quantity here rather than the standard deviations of the two rounds.

QB 5.17 Using data/attendance-grades.csv, write R code to compare cumulative GPA between students above and below the median attendance. Report the interval, then write a short paragraph stating precisely what your result establishes and what it does not.

Paired designs

QB 5.18 A paired \(t\)-test on 15 pairs has how many degrees of freedom?

  1. 13
  2. 14
  3. 28
  4. 30

QB 5.19 The paired \(t\)-test is equivalent to

  1. a two-sample \(t\)-test with var.equal = TRUE
  2. a one-sample \(t\)-test on the differences
  3. Welch’s test with adjusted degrees of freedom
  4. a \(Z\)-test on the difference of means

QB 5.20 Pairing gives the largest gain in precision when

  1. the two sample sizes are equal
  2. the two columns are strongly correlated
  3. the two columns are independent
  4. the sample is large

QB 5.21 Which analysis is appropriate? (match each design to a test)

  1. Yields on 20 plots under seed A and on 20 different plots under seed B
  2. Yields on the same 20 plots in two consecutive seasons, one seed each
  3. Blood pressure of 40 patients, half given a drug and half a placebo
  4. Blood pressure of 40 patients, each measured before and after a drug
  1. Two-sample \(t\)-test
  2. Paired \(t\)-test

QB 5.22 A researcher analyses genuinely paired data as two independent samples. The most likely consequence is

  1. a \(p\)-value that is too small
  2. a \(p\)-value that is too large
  3. a confidence interval that is too narrow
  4. no change, since the sample means are the same either way

Paired reasoning

QB 5.23 Explain why the paired test uses the standard deviation of the differences rather than the standard deviations of the two columns, and what this achieves.

QB 5.24 A colleague argues that pairing should always be used because it is “more powerful”. Under what circumstances is that claim false, and what is the cost of pairing when the two columns are unrelated?

QB 5.25 A study reports before-and-after measurements on the same 12 villages and finds a significant improvement. A critic replies that there was no comparison group, so nothing has been shown. Explain what the paired test does establish here and what the critic is right about.

Paired numerical

QB 5.26 Nine households report monthly food expenditure, in rupees, before and after a ration card reform.

Household Before After
1 4200 4050
2 3800 3750
3 5100 4800
4 4600 4500
5 3900 3700
6 4400 4250
7 5000 4650
8 4100 4050
9 4700 4400
  1. Compute the differences, \(\bar{d}\) and \(s_d\).
  2. Test at the 5% level whether expenditure changed.
  3. Construct a 95% confidence interval for the mean change.
  4. State one reason the change should not be attributed to the reform.

QB 5.27 A paired study on \(n = 16\) pairs gives \(\bar{d} = 3.5\) and \(s_d = 6.4\).

  1. Compute the test statistic and its degrees of freedom.
  2. Test at the 5% and 1% levels.
  3. How large would \(n\) have to be for the same \(\bar{d}\) and \(s_d\) to reject at the 1% level?

QB 5.28 Two analyses of the same twelve paired observations report \(s_d = 2.1\) and, from the two columns separately, \(s_x = 9.4\) and \(s_y = 9.1\).

  1. Compute the standard error the paired test would use.
  2. Compute the standard error an unpaired test would use.
  3. Explain the ratio between them in terms of what pairing removes.

QB 5.29 Using data/student-survey-sleep.csv, write R code that deliberately performs a paired test on two variables that are not paired in any meaningful sense. Report what R does, and explain why the output is meaningless even though no error is raised.