# PoSWW 32: Confidence Trick

We are so, so sick of Specialist Mathematics. But, it’s gotta be done.

A couple of blog readers have emailed and suggested that we have given insufficient time to the question below, from 2022 Specialist Exam 1. They are absolutely correct.

To state it as clearly as is humanly possible, the third sentence in part (b) is not ambiguous. The sentence is a clear statement of fact and students have every right, and arguably a responsibility, to answer the question on the basis of the plain meaning of the statement. Any argued implausibility in reference to the question stem is entirely irrelevant.

## 19 Replies to “PoSWW 32: Confidence Trick”

1. Sai says:

Considering that there have been three major prob&stats mistakes across the Specialist Exams, I’d guess the title is to indicate VCAA’s hat trick. Man, this one is really bad, because if you let C be the random variable for the new time it takes for a coffee machine to dispense a cup of coffee, then the statement in error would be interpreted as . There just is no way of being charitable to the writer of this question. Those are some fast coffee machines though!

1. marty says:

Hands up everyone who is in a mood to be charitable?

1. Red Five says:

You’re right Marty in that the 9 seconds, as the question is written is the mean time for the total 25 cups, not 9 seconds per cup.

I do wonder though if the “solutions” will need to be re-written once the assessors read them.

2. Terry Mills says:

I agree with Marty. It is a standard question about finding a confidence interval for a population mean, as I hinted at earlier.

My only quibble is that we are not told that the sample is a random sample, and, reading the question, it is unlikely to be so. Statistical analysis is based on random samples. To me, this is like not mentioning that a function is continuous when continuity is required for a proof.

In practice, how would one try to make the sample random? One idea would be to choose 10 times at random for the rest of the day/shift and measure the time taken for those times.

1. Red Five says:

There are many ways to make a sample random. Unfortunately neither Specialist nor Methods gets close to a proper understanding of what “sample” actually means. “Random” is even more problematic and the study designs do not even attempt a definition or explanation of concept here as far as I can see.

1. Terry Mills says:

As I have written before, many university text books do not present a correct definition of a random sample. I have a list.

My suggestion above was only one way to make the sample appear to be random.

3. John (No) Friend (of VCAA) says:

Holy cow! That machine sure did need a service.

Even if VCAA does intend exactly what it wrote (which I don’t think it does, but you can only eat what’s put on your plate), there’s a snag in calculating the CI that I doubt students will have met in any of the textbook questions or seen in a class.

The initial calculations of the CI are so trivial that I doubt anyone has considered them. I encourage you to do so. And then to answer the question:

What’s the snag?

4. Red Five says:

I assume you’re suggesting that per cup, to one decimal place the CI is which is indeed a problem that suggests VCAA does not want students to interpret the question AS IT IS WRITTEN.

WiTCH indeed!

1. John Friend says:

sd = 1.5, z = 2 (say) and n = 25.
Now, if sample mean = 9/25 ….

1. SRK says:

Is your point that the interval includes negative values? Maybe the servicing was so good that a customer now gets their coffee even before the barista has put the grind in the dispenser.

1. John Friend says:

Kapow!

So what do we do? Keep the negative value in our confidence interval? (Not withstanding your excellent point of “so good” servicing). What would students do? What should they be expected to do?

How many different answers should VCAA accept because of this debacle?
(And I don’t want to hear anyone say it’s no use crying over spilt coffee) .

2. Red Five says:

I really need to sleep before trying to multiply and add decimals in my head…

1. John Friend says:

RF, better that you’re tired than I’m loco!! (And I was ready to believe the latter).

2. Sai says:

Perhaps 25 cups of coffee in 9 seconds would do the trick.

1. Red Five says:

Well played. Well played indeed!

A teacher’s life is far from normally distributed at the moment…

2. John Friend says:

Oh, that’s very, very good! I must be tired too, not to have thought of it.

1. Red Five says:

It is peak of exam/report season. All is forgiven.

5. John (No) Friend (of VCAA) says:

It occurs to me that if one wishes to argue that

“Following the service, the mean time taken to dispense 25 cups of coffee is found to be 9 seconds”

‘obviously’ means that the mean time for dispensing one cup of coffee is 9 seconds, one might first reflect on:

“Following the service, the mean time taken to dispense 25 cups of coffee is found to be 200 seconds”

and argue why it is (or isn’t!) ‘obvious’ that this means that the mean time for dispensing one cup of coffee is 200 (*) seconds ….

* Or 100 seconds. Or 50 seconds. Or 300 seconds. Where’s the line in the sand for (inevitably) switching sides in the argument ….?

1. John (No) Friend (of VCAA) says:

The * should also be next to the 200 in the alternate sentence.

The point is that you can’t let a number (9, 200, 100, 300, whatever) dictate the unequivocal meaning of the sentence.

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