WitCH 88: Unwarranted Confidence

One more from the 2022 Specialist Mathematics Exam 2. Belatedly, we’ve decided this one deserves its own post. It’s probably more of a PoSWW. But, for those weirdos who like to think about this stuff, there are probably also aspects to discuss.

40 Replies to “WitCH 88: Unwarranted Confidence”

  1. I wonder what VCAA thinks the purpose of a confidence interval actually is… have they ever publicly said anything about it?

  2. Answer F: (2.6,2.6)

    Since the MEAN IS KNOWN, we can be 100% confident and do not need a sample to tell us anything.

      1. An excellent point. I guess I never think about confidence intervals ever being closed intervals.

        But then neither do VCAA, so probably not much to be gained.

  3. It seems to me that the author of the question does not understand confidence intervals. This sounds harsh, but the error in framing the question is a fundamental error in understanding rather than a slip in calculation or sloppy wording.

    Many mathematics teachers are not well versed in statistics. Perhaps some of them move on to setting VCE examinations – and we see the result in questions like this.

    (A story as I recall it.) A.D. Hope was a famous Australian poet. His first degree was in psychology – which is not called “rats and stats” for nothing! Anyway, he had a job lecturing on statistics at the Sydney Teachers College to pre-service teachers. He tells the story how one student in a class said “Mr Hope, what you have written on the board is wrong”. Hope asked “How do you know?” The student said “I have an honours degree in pure mathematics.” Hope invited the student to write the correct stuff on the board and the student obliged. Hope then invited the student to give the rest of the lectures on statistics: Hope wrote that he learned a great deal from these lectures!

    1. But Terry. I quote VCAA from the Age article:

      “Our exams are set by a panel of highly experienced teachers and academics and reviewed by specialists in each field of study.”

      1. That is depressing, that these errors slip by ‘specialists’ but get caught by a ragtag collection of commenters online. Did each specialist see no issue in asking for a confidence interval when the population parameters were known?

          1. Terry, that’s a reasonable suggestion. Maybe (probably) that \displaystyle is what the writers meant. However, this suggestion is also fundamentally flawed (*).

            The random variable \displaystyle \overline{C} follows a normal distribution:

            Normal\displaystyle \left( \mu = 2.6, \, \, \sigma = \frac{0.3 \times 0.1}{10} = 0.003\right) (**)

            If the values of a and b such that \displaystyle Pr \left( a < \overline{C} < b\right) were what the writers intended, then we have the PoSWW (Proof of Stupidity Without Words) discussed here:

            PoSWW 20: Unconventional Wisdom

            The only way of rehabilitating Question 19 is to delete it, appoint a new and competent writer and start over.

            * OK, it’s not \displaystyle quite flawed because the question asks for \displaystyle a interval, not \displaystyle the interval. But I’d argue:
            1) The only reason that \displaystyle a interval is asked for is because the ‘95% confidence interval’ is only approximate.
            2) Therefore the underlying thinking behind the question (if this is what the writers intended) is undoubtedly flawed as is discussed at the link above.
            At the very least, the re-imagining is guilty of \displaystyle promoting flawed thinking.

            ** The actual values of the parameters of the normal distribution are not important for what follows. The details of their calculation are available if there is sufficient curiosity.

        1. I take great exception to “ragtag”. I prefer motley crew.

          (Besides, I refuse to join any club that would have me as a member [as a great man once said])

          And, as with all things, it all depends on your definitions. The definition of “specialist” (and “academic”) in this case.

    2. Terry, I totally agree that the question demonstrates a fundamental lack of understanding of confidence intervals. This is not a harsh conclusion at all.

      I think the writer was far more interested in being ‘creative and novel’ than in being correct. And their desire for the former far outreached their ability for the latter.

  4. Not being a statistician, I have also been caught out by a fundamental misunderstanding of confidence intervals. But that is why I neither teach, nor set exams in, statistics. Anyway, I do know just enough to recognize that what the examiners are mistakenly calling a confidence interval is nothing of the kind.

    There’s a meta-concern here, not just about confidence intervals. We have seen similar – and equally egregious – misunderstandings occur again and again. The students who do best, then, are those who can work within the VCAA system of incorrect definitions, half-truths (at best), misapplied techniques, and general woolly muddle-headedness. These students then stagger into tertiary mathematics with heads so full of rubbish that teaching them is far harder than it should be.

      1. On this point I take (small) issue…

        If universities are not going to require students to study Specialist Mathematics for entry to courses, can they then complain that the Mathematics students learn is crap?

        OK, I will concede that those setting the admissions rules are a world away from those doing the teaching of the students being admitted, but I do think there is something of a correlation between the lowering of entry requirements and the lowering of the quality of VCE examinations.

        More than happy to be proven wrong, statistically…

          1. Or wine …?

            I know that \displaystyle some mathematicians have attempted constructive action. Perhaps their direction action has come a little too late to achieve immediate results, but nevertheless …

            And there are a very small number of mathematicians (of which you are one, Marty, and please don’t say that you’re no longer a mathematician) that have tried for many years.

            There are far more mathematics teachers than mathematicians – if mathematicians are to blame (and yes, they certainly \displaystyle share blame), they can take a number and stand in line behind the teachers.

            1. Nah. As a class, mathematicians don’t give a stuff, and haven’t for decades. They are more responsible than anyone else for the obscenity that school mathematics education has become.

              1. I’m not blaming Mathematicians. I’m blaming the university bureaucrats who think changing entry requirements for courses will have the effect only of increasing their profits.

                Mathematicians, or those who lecture at universities, have to wear the consequences of these decisions in no small part.

                  1. And since I have no doubt your sample size is much larger than mine (someone really needs to make a T-shirt out of that…) I will leave my side of the argument where it is.

                    The other side of my argument (of which you are again free to tear strips away…) is that, on the whole, I do not blame teachers either.

                    Teachers are paid to do their job. A lot of the job description is “do what the principal says”. In spite of what Mark Latham has recently claimed, teachers do not have secure jobs and rocking the boat is not something many principals like to see.

                    Apparently, it is worse in NSW because the teacher registration board north of the border is worse than VIT (yep, I’ll believe that when I see it… not doubting, just find it hard to fathom…) and so there is even more pressure to “stay inside the lines.”

                    1. With the reported shortage of teachers (over 2,000 unfilled positions in NSW: https://www.theguardian.com/australia-news/2022/oct/11/teacher-vacancies-hit-2000-across-nsw-as-some-schools-record-14-unfilled-roles#:~:text=The%20NSW%20Teacher%20Supply%20Strategy,offered%20better%20pay%20and%20conditions.), I am astounded that teachers still feel too insecure to rock the boat, even a little bit.

                      Principals should be \displaystyle encouraging their teachers to take a stand against mediocre and second rate curricula, incompetently written examinations, deceitful Examination Reports etc.

                      But I think many Principals are grappling with the dichotomy of legacy thinking (“If you don’t like it here, you know where the door is.”) versus the new reality (“Yes, I know where the door is. Good luck replacing me – like for like – when I walk through it.”)

                      The ability of Principals to adopt a new attitudinal paradigm will play a large part in how many teachers stay in the Public system and how many leave it for the Private system.

                      There has never been a better time to rock the boat and demand better standards and accountability. Rocking the boat would also be a lot easier if some of the professional bodies claiming to represent teachers stopped being Government stooges.

                      As long as teachers remain silent and accept sub-mediocrity – the silence of the lambs – they carry a huge amount of blame.

  5. About the titles for the trifecta of errors on confidence intervals – I should apologise for having used the best title some time ago: “A Lack of Confidence”.
    Since then, I think we can all feel an even greater level of a lack of confidence in VCAA exam questions.

  6. I’ve made the following comment elsewhere (https://mathematicalcrap.com/2022/11/07/secret-2022-specialist-business-exam-2-discussion/), but I’d really like to repeat it here to make the conceptual flaw crystal clear in this dedicated post to the question:

    “There is no doubt that the question is defective and so cannot be answered.

    A confidence interval is based on a given set of sample data. It gives an estimated range of values that is likely to include an unknown population parameter (such as population mean) with a certain level of ‘confidence’ (*).

    The question does not give a sample mean. [In fact, the question is asking] that a ‘confidence interval’ based on the population mean … be calculated for a sample mean. That’s a load of bollocks.

    Furthermore, the given information enables the population mean to be \displaystyle exactly calculated, which makes finding a confidence interval for the population mean pointless [anyway] (assuming the sample mean \displaystyle was given in the first place so that such a calculation could actually be done).

    * More can be said about the meaning of the level of ‘confidence’, but it’s not relevant.”

    1. The 95% confidence interval for the population mean was not asked for, that explains why the sample mean was not given.

      1. Anonymous, I encourage you to object to people’s objections. But it would be helpful to say things in some detail. Your comments aren’t particularly enlightening.

        1. Given the parameters of the population, the question asked for an interval that one would expect to find the mean of a sample at 95% confidence level.

          1. I still have no idea what you’re saying, or why. I’ll go back to my default position, that this is stats crap, and I don’t care. If others want to try to figure out you’re point, that’s fine.

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