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

Unbiased estimates, validity and reliability Problem set B Worked solutions

Skills

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

    1. (a) (a) 'Which film is the best ever made?'

      • 'Best' is a matter of taste with no agreed measure, and the choice changes with mood.

    2. (b) (b) 'How many bedrooms are there in your home?'

      • It asks for a countable fact, so it measures what is intended, and the answer would not change if repeated.

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

    1. (a) (a) 26, 31, 18, 35, 29, 22

      • From the calculator: 1-Variable Stats on 26, 31, 18, 35, 29, 22
      • \(\bar x=26.8\), \(s_{n-1}=6.178\)
      • \(s_{n-1}^2=38.2\)

    2. (b) (b) −2, 7, −9, 4, 1, −5, 3

      • From the calculator: 1-Variable Stats on −2, 7, −9, 4, 1, −5, 3
      • \(\bar x=−0.143\), \(s_{n-1}=5.551\)
      • \(s_{n-1}^2=30.8\)

  3. The masses of 9 parcels have standard deviation \(s_n=0.84\) kg, found by dividing by n. Find the unbiased estimate \(s_{n-1}\) of the population standard deviation.

    • \(s_{n-1}=\sqrt{\dfrac{n}{n-1}}\,s_n\)
    • \(=\sqrt{\dfrac{9}{8}}\times 0.84\)
    • \(=0.891\)

  4. A sample's unbiased standard deviation \(s_{n-1}\) is 1.0247 times its \(s_n\). Find the sample size n.

    • \(\dfrac{n}{n-1}=1.0247^2=1.0500\)
    • \(n=1.0500(n-1)\), so \(n=21.00\)
    • \(n=21\)

  5. In six races a swimmer's lengths-per-minute were 21, 17, 24, 18, 26 and 20. These are a random sample of her results.

    1. (a) Find \(\sum(x-\bar x)^2\) and hence the unbiased estimate of the variance of her lengths per minute.

      • \(\bar x=21\)
      • Deviations: 0, −4, 3, −3, 5, −1. \(\sum(x-\bar x)^2=0+16+9+9+25+1=60\)
      • \(s_{n-1}^2=\dfrac{60}{5}=12\)

    2. (b) For a different sample of 8 races, \(\sum x=96\) and \(\sum x^2=1210\). Find the unbiased estimate of the variance.

      • \(\sum(x-\bar x)^2=1210-\dfrac{96^2}{8}=58\)
      • \(s_{n-1}^2=\dfrac{58}{7}=8.29\)

In context

  1. A park ranger catches trout to estimate the mean length of all trout in a region. The lengths (cm) of eight trout are 31.2, 28.5, 35.0, 30.1, 33.4, 29.8, 32.6, 30.9.

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

      • From the calculator: 1-Variable Stats
      • \(\bar x=31.4\)

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

      • \(s_{n-1}=2.116\)
      • \(s_{n-1}^2=4.48\)

    3. (c) The ranger caught all eight trout in one pond. Explain why the estimates may not be valid.

      • All the trout came from one pond, so the sample may not represent trout in the whole region: the estimates may not be valid.

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

      • Catch a second random sample from other ponds (or measure again) and check that the estimate is close to the first.

  2. A leisure centre leaves a questionnaire at the pool entrance: 'How many times did you swim at the centre last month?' After one week the responses were 9, 14, 6, 11, 10, 640.

    1. (a) Explain why it is reasonable to remove the value 640.

      • 640 swims in a month is over 20 a day: impossible, so it is an error.

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

      • Remove 640. From the calculator: \(\bar x=10.0\), \(s_{n-1}=2.915\)
      • \(s_{n-1}^2=8.50\)

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

      • 'Swim' is not defined (does a short dip count?) and people may not remember, so the question is not valid.

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

      • Only people who came in, and 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: regular swimmers are at the pool 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 swimmers, and check that the estimates agree.

  3. A teacher gives the students two different versions of a vocabulary test, each with 30 questions on the same topics, on the same day, and compares the two sets of scores.

    1. (a) Define reliability.

      • Reliability is the consistency of results: measuring again in the same way gives similar results.

    2. (b) Name the reliability test used.

      • Parallel forms

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

      • Test-retest: give the same test again later and compare. Or internal consistency: check that questions on the same topic agree with each other.

  4. A tutor asks at the end of a course: 'Do you think this course will improve your exam grade?'

    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 the grade; it is a leading question, so people who have just finished tend to say yes; 'improve' 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 tutor later compares the answers with an independent measure, the students' real exam grades. What aspect of validity is being tested by this comparison?

      • Criterion-related validity: the tutor compares the answers with an independent measure, the real grades.

  5. A calculator's 1-Variable Stats for the heights of 9 randomly chosen seedlings shows \(\sigma_x=0.620\) cm and \(s_x=0.658\) cm.

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

      • \(s_x\) divides by \(n-1\); \(\sigma_x\) 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.

      • \(s_{n-1}^2=0.658^2=0.433\)
      • Check: \(\dfrac{9}{8}\times0.620^2=0.432\), the same to rounding.

Stretch

  1. The masses (kg) of 10 suitcases are summarised by \(\sum x=130\) and \(\sum x^2=1790\). The raw list is lost.

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

      • \(\bar x=\dfrac{130}{10}=13.0\)

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

      • \(s_{n-1}^2=\dfrac{\sum x^2-\frac{(\sum x)^2}{n}}{n-1}=\dfrac{1790-1690.0}{9}\)
      • \(=11.1\)

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