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IB Mathematics: Applications and Interpretation HL · AHL 8: Statistics 2 · AHL 8.10

Unbiased estimates, validity and reliability Problem set A Worked solutions

Skills

  1. Say whether each survey question is valid and whether its results are reliable. Justify each answer.

    1. (a) (a) 'Which is the most beautiful city in Europe?'

      • Beauty is an opinion with no measure, so it does not measure anything definite. People would also give different answers on different days.

    2. (b) (b) 'How many pets live in your home?'

      • It asks for a countable fact, so it measures what it is meant to, and the same person gives the same answer each time.

  2. Find unbiased estimates of the population mean and variance from each sample. Give each to 3 significant figures.

    1. (a) (a) 14, 19, 8, 23, 17, 11

      • From the calculator: 1-Variable Stats on 14, 19, 8, 23, 17, 11
      • \(\bar x=15.3\), \(s_{n-1}=5.465\)
      • \(s_{n-1}^2=29.9\)

    2. (b) (b) 5, −3, 8, −6, 2, 0, −4

      • From the calculator: 1-Variable Stats on 5, −3, 8, −6, 2, 0, −4
      • \(\bar x=0.286\), \(s_{n-1}=5.057\)
      • \(s_{n-1}^2=25.6\)

  3. The heights of 14 saplings have standard deviation \(s_n=6.3\) cm, found by dividing by n. Find the unbiased estimate \(s_{n-1}\) of the population standard deviation.

    • \(s_{n-1}=\sqrt{\dfrac{n}{n-1}}\,s_n\)
    • \(=\sqrt{\dfrac{14}{13}}\times 6.3\)
    • \(=6.54\)

  4. For a sample, \(s_{n-1}\) is 1.0328 times \(s_n\). Find the sample size n.

    • \(\dfrac{n}{n-1}=1.0328^2=1.0667\)
    • \(n=1.0667(n-1)\), so \(n=16.00\)
    • \(n=16\)

  5. In six matches a team scored 14, 10, 17, 12, 9 and 16 goals. These are a random sample of its scores.

    1. (a) Find \(\sum(x-\bar x)^2\) and hence the unbiased estimate of the variance of the number of goals per match.

      • \(\bar x=13\)
      • Deviations: 1, −3, 4, −1, −4, 3. \(\sum(x-\bar x)^2=1+9+16+1+16+9=52\)
      • \(s_{n-1}^2=\dfrac{52}{5}=10.4\)

    2. (b) For a different sample of 10 matches, \(\sum x=120\) and \(\sum x^2=1500\). Find the unbiased estimate of the variance.

      • \(\sum(x-\bar x)^2=1500-\dfrac{120^2}{10}=60\)
      • \(s_{n-1}^2=\dfrac{60}{9}=6.67\)

In context

  1. Karim weighs bags of flour from one supermarket to estimate the mean mass of all bags sold in his town. The masses (g) of seven bags are 1004, 998, 1010, 995, 1002, 1008, 1001.

    1. (a) Find an unbiased estimate of the population mean.

      • From the calculator: 1-Variable Stats
      • \(\bar x=1002.57\ldots=1003\) g (nearest gram)

    2. (b) Find an unbiased estimate of the population variance.

      • \(s_{n-1}=5.287\)
      • \(s_{n-1}^2=28.0\)

    3. (c) Karim took all seven bags from one supermarket. Explain why the estimates may not be valid.

      • Karim used only one supermarket, so the sample may not represent all the bags sold in the town: the estimates may not be valid.

    4. (d) How could the reliability of the result in part (a) be tested?

      • Take a second random sample (or repeat the weighing) and check that the estimate is close to the first.

  2. A bus company leaves a questionnaire on its buses for passengers to post back: 'How many journeys did you make on our buses last month?' After one week the responses were 22, 15, 30, 18, 25, 1350.

    1. (a) Explain why it is reasonable to remove the value 1350.

      • 1350 journeys in a month is over 40 a day: impossible, so it is an error.

    2. (b) For the remaining values, find unbiased estimates of the population mean and variance.

      • Remove 1350. From the calculator: \(\bar x=22.0\), \(s_{n-1}=5.874\)
      • \(s_{n-1}^2=34.5\)

    3. (c) Comment on the content validity of the question.

      • 'Journey' is not defined (is a return trip one or two?) and people may not remember exactly, so the question is not valid.

    4. (d) Comment on the validity of the sampling method.

      • Only bus passengers who chose to post it back replied: not random, so the sample is biased and not valid.

    5. (e) Is the estimate of the mean more likely to be an over-estimate or an under-estimate? Justify.

      • Over-estimate: people who travel often are on the buses more and more likely to reply.

    6. (f) How could the reliability of the survey be improved?

      • Repeat the survey in other weeks and with more passengers, and check that the estimates agree.

  3. A psychologist gives the same 20-question personality test to a group of students, then gives them the same test again two weeks later and compares the two sets of scores.

    1. (a) Define reliability.

      • Reliability is the consistency of results: repeating the test in the same conditions gives similar results.

    2. (b) Name the reliability test used.

      • Test-retest

    3. (c) Suggest a different way to assess the reliability of the test.

      • Parallel forms: give two different versions of the test and compare. Or internal consistency: check that questions on the same trait agree with each other.

  4. A fitness trainer asks at the end of a programme: 'Do you think this programme will improve your fitness?'

    1. (a) Define validity.

      • Validity is how well a test or question measures what it is meant to measure.

    2. (b) Give two reasons why this question does not have content validity.

      • It asks for an opinion about the future, not a measure of fitness; it is a leading question, so people who have just finished tend to say yes; 'fitness' is not defined.

    3. (c) Explain why making the questionnaire anonymous might improve its validity.

      • People are more likely to answer honestly if no one can trace the answer to them, so the answers are closer to what they really think.

    4. (d) The trainer later compares the answers with an independent measure of fitness, such as a fitness test. What aspect of validity is being tested by this comparison?

      • Criterion-related validity: the trainer compares the answers with an independent measure of fitness.

  5. A calculator's 1-Variable Stats for the times of 8 randomly chosen runners shows \(\sigma_x=1.84\) s and \(s_x=1.97\) s.

    1. (a) Which of the two values is used to estimate the standard deviation of all runners? Give a reason.

      • \(s_x\) divides by \(n-1\); \(\sigma_x\) divides by n and describes the sample only.

    2. (b) Explain why the standard deviation that divides by n tends to be too small as an estimate of the population value.

      • The sum of squared deviations is smallest about \(\bar x\), so it is too small about \(\mu\).
      • Dividing by \(n-1\), not n, corrects this on average.

    3. (c) Find the unbiased estimate of the population variance.

      • \(s_{n-1}^2=1.97^2=3.88\)
      • Check: \(\dfrac{8}{7}\times1.84^2=3.87\), the same to rounding.

Stretch

  1. The times (seconds) of 8 pit stops are summarised by \(\sum x=96\) and \(\sum x^2=1260\). The raw list is lost.

    1. (a) Find an unbiased estimate of the population mean.

      • \(\bar x=\dfrac{96}{8}=12.0\)

    2. (b) Find an unbiased estimate of the population variance.

      • \(s_{n-1}^2=\dfrac{\sum x^2-\frac{(\sum x)^2}{n}}{n-1}=\dfrac{1260-1152.0}{7}\)
      • \(=15.4\)

jmaths.xyz · AI HL · AHL 8.10 Unbiased estimates of the mean and variance; validity and reliability