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March 29, 2024, 04:37:55 pm

Author Topic: VCE Methods Question Thread!  (Read 4803157 times)  Share 

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svnflower

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Re: VCE Methods Question Thread!
« Reply #18720 on: August 22, 2020, 09:18:53 pm »
0
Hello :)

How would I solve this question? (attached below)




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Re: VCE Methods Question Thread!
« Reply #18721 on: August 22, 2020, 09:28:39 pm »
+2
Hey there!

Recall that for some integral \(\int \ (ax+b)^n \ dx\) we have the anti-derivative being \(\frac{(ax+b)^{n+1}}{a(n+1)} + C\). Try applying this to your question by rearranging the given integral so that it imitates an integral of this form. Hints in the spoiler. Hope this helps :)
 
Spoiler

Spoiler
Apply with a = -2, b = 9, n = 1/2
Spoiler
Answer should be 26/3! :)
Spoiler
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eloisegrace

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Re: VCE Methods Question Thread!
« Reply #18722 on: August 23, 2020, 10:30:30 am »
0
Hello!

I have a question from Cambridge 16D q9. It is regarding the normal distribution and my attempts kinda got me nowhere...

The volume of soft drink in a 1-litre bottle is normally distributed. The soft drink company needs to calibrate its filling machine. They don’t want to put too much soft drink into each bottle, as it adds to their expense. However, they know they will be fined if more than 2% of bottles are more than 2 millilitres under volume. The standard deviation of the volume dispensed by the filling machine is 2.5 millilitres. What should they choose as the target volume (i.e. the mean of the distribution)? Give your answer to the nearest millilitre.

The answer is 1004mL btw :) Thanks!
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Re: VCE Methods Question Thread!
« Reply #18723 on: August 23, 2020, 10:41:46 am »
+2
Hello!

I have a question from Cambridge 16D q9. It is regarding the normal distribution and my attempts kinda got me nowhere...

The volume of soft drink in a 1-litre bottle is normally distributed. The soft drink company needs to calibrate its filling machine. They don’t want to put too much soft drink into each bottle, as it adds to their expense. However, they know they will be fined if more than 2% of bottles are more than 2 millilitres under volume. The standard deviation of the volume dispensed by the filling machine is 2.5 millilitres. What should they choose as the target volume (i.e. the mean of the distribution)? Give your answer to the nearest millilitre.

The answer is 1004mL btw :) Thanks!

I'd approach this by first asking "how many standard deviations away from the mean do I need to be to have only 2% of values be lower?".
Then you can you go, okay, the mean needs to be x standard deviations above 998 mL.
Thus the target volume will be 998 [the 2% threshold] + x [the number of standard deviations] * 2.5 [the size of each standard deviation].

Let me know if this clears things up or not :)

lm21074

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Re: VCE Methods Question Thread!
« Reply #18724 on: August 23, 2020, 10:48:56 am »
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Hi!
This is probably a very simple question to answer but with the formula below, I'm not entirely sure why you have to divide by r!. I think it's in order to avoid repeats, but how does dividing by r! do this?

Thanks so much! :)
« Last Edit: August 23, 2020, 10:52:27 am by lm21074 »
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Re: VCE Methods Question Thread!
« Reply #18725 on: August 23, 2020, 11:28:06 am »
+5
Hi!
This is probably a very simple question to answer but with the formula below, I'm not entirely sure why you have to divide by r!. I think it's in order to avoid repeats, but how does dividing by r! do this?

Thanks so much! :)

You should try and give more context when asking questions like this - it can often be confusing pulling out equations out of nowhere like this. However, I think you're talking about comparing permutations to combinations? In which case, the formula for the number of permutations of r objects in a set of n would be:



Whereas the the formula for the number of r COMBINATIONS in a set of n would be:



And you want to know why combinations require dividing by r! - is that right?

In this instance, you need to ask not just what's different between the equations, but between the actual situation. A combination is easier to understand because we can put it in terms of real world understanding. For example - let's say you're making a 3 veggie salad out of lettuce, cucumber, carrot, and red onion. Well, if you were to make the salad by adding lettuce, then cucumber, then carrot, that would be the same as the salad made by adding cucumber, then lettuce, then carrot - the ingredients are the same, after all! However, a permutation is a bit trickier to think about - but you can think of it as the recipe. Although the final salad is the same, the actual recipe is different - the order that things are added is important. I think this is a point you might already understand, so I won't dwell on it more than this.

Okay, so what does this have to do with the formulas? Well, we know from the formula for permutations, there are a total of:



different recipes. But we don't have nearly that many salads, do we? Well, if you trace out all the combinations by writing them out, you'd find there are only 4 different salads we could make - why is there so many different ways to make them?? Well, let's consider the lettuce, carrot, cucumber recipe. We could make it in the following orders:

Lettuce, carrot, cucumber
Lettuce, cucumber, carrot
Cucumber, lettuce, carrot
Cucumber, carrot, lettuce
Carrot, lettuce, cucumber
Carrot, cucumber, lettuce

6 ways in total. Okay, but we only want one of these to be preserved, so we should subtract 6 from first equation! ... But if we did that, we'd lose ALL of the combinations. Okay, so maybe instead we subtract 5. But we need to do this for all the other permutations, too - which would require knowing how many combinations there are to begin with! But, here's a thought. What if we were making this salad, and our list never included red onion? Well, we'd have 6 different recipes we could follow, but now there's only possible salad - because we're choosing 3 ingredients from a list of 3 ingredients - every time, we're just going to choose all of them. In this case, how can we get from 6 permutations to 1 combination? We could subtract 5, of course - but we could also divide by 6, which is just the total number of permutations of the full list. Think about it - if we divide each of our permutations from the set into groups of ALL POSSIBLE permutations, then all that's going to be left is one group of permutations.

Okay, so let's go back to our 4 ingredient for 3 veggie salad problem. In this case, there are 24 different recipes we could follow. We know that the total amount of possible permutations for a recipe of 3 ingredients is going to be 3!. So, if we take all 24 different recipes, and divide them into 3! (6) groups, one of these groups should have just one final combination of recipes. Firstly, here's all 24 different recipes:

Spoiler
Lettuce, carrot, cucumber
Lettuce, cucumber, carrot
Cucumber, lettuce, carrot
Cucumber, carrot, lettuce
Carrot, lettuce, cucumber
Carrot, cucumber, lettuce

Red onion, carrot, cucumber
Red onion, cucumber, carrot
Cucumber, red onion, carrot
Cucumber, carrot, red onion
Carrot, red onion, cucumber
Carrot, cucumber, red onion

Red onion, lettuce, cucumber
Red onion, cucumber, lettuce
Cucumber, red onion, lettuce
Cucumber, lettuce, red onion
Lettuce, red onion, cucumber
Lettuce, cucumber, red onion

Red onion, lettuce, carrot
Red onion, carrot, lettuce
Carrot, red onion, lettuce
Carrot, lettuce, red onion
Lettuce, red onion, carrot
Lettuce, carrot, red onion

Now, if we divide them into six groups... That's the equivalent of just taking the first set from each group of permutations, which gives us 4 possible salads in total.

Hopefully that makes a bit more sense - the idea is if we divide by r!, well r! is the number of permutations we could be having, so it's like dividing every permutation into their own combination groups, and just picking out one of the combinations from each permutation group. It's hard to think about abstractly, but hopefully makes more sense now that you've seen it as an example.

eloisegrace

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Re: VCE Methods Question Thread!
« Reply #18726 on: August 23, 2020, 12:35:05 pm »
0
I'd approach this by first asking "how many standard deviations away from the mean do I need to be to have only 2% of values be lower?".
Then you can you go, okay, the mean needs to be x standard deviations above 998 mL.
Thus the target volume will be 998 [the 2% threshold] + x [the number of standard deviations] * 2.5 [the size of each standard deviation].

Let me know if this clears things up or not :)
Hi Bri MT,
Thanks, I now understand the method. I got the answer as 1003.134 so why does this round up to 1004mL?
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Sine

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Re: VCE Methods Question Thread!
« Reply #18727 on: August 23, 2020, 01:22:13 pm »
+4
Hi Bri MT,
Thanks, I now understand the method. I got the answer as 1003.134 so why does this round up to 1004mL?
Assuming all your calculations are correct maybe the machine can only be adjusted to the nearest 1mL. EDIT:  just saw your answer should be to the nearest mL. So if you go any low than the exact value you will have more than 2% of bottles under 998mL. So then you need to adjust it to 1004mL where there will be even less than 2%. It is less about "rounding" the number but thinking about the result in the context of the question.

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Re: VCE Methods Question Thread!
« Reply #18728 on: August 23, 2020, 03:44:40 pm »
+2
Assuming all your calculations are correct maybe the machine can only be adjusted to the nearest 1mL. EDIT:  just saw your answer should be to the nearest mL. So if you go any low than the exact value you will have more than 2% of bottles under 998mL. So then you need to adjust it to 1004mL where there will be even less than 2%. It is less about "rounding" the number but thinking about the result in the context of the question.

Not a criticism of your reply - but it would be nice to get some clarification from VCAA about this issue. The question does not state that the volumes in the bottle are restricted to whole numbers of mL, nor does it state that the final rounded answer should be such that less than 2% of bottles are below the allowed volume. Getting an answer of 1003.3 and rounding to the nearest mL should give 1003.

The other reasonable thing to do would be to accept both answers - which has occurred in the past.

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Re: VCE Methods Question Thread!
« Reply #18729 on: August 23, 2020, 04:10:28 pm »
+3
Not a criticism of your reply - but it would be nice to get some clarification from VCAA about this issue. The question does not state that the volumes in the bottle are restricted to whole numbers of mL, nor does it state that the final rounded answer should be such that less than 2% of bottles are below the allowed volume. Getting an answer of 1003.3 and rounding to the nearest mL should give 1003.

The other reasonable thing to do would be to accept both answers - which has occurred in the past.
Yeah, I definitely understand why someone would round down to 1003mL. They should be more clear in the question. I.e. the final rounded answer should result in less than 2% below 998mL. I think the best thing to do if there is any ambiguity is to write a statement at the end justifying why you went to 1003 or 1004. If you can show the understanding I think you should get the mark either way.

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Re: VCE Methods Question Thread!
« Reply #18730 on: August 24, 2020, 09:44:32 pm »
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Yes - as long as pages aren't easily removed and the book has a single spine, you're fine. Hell, there have been people in the past who have just taken their textbook, duct-taped it to an A4 book they took notes in all year, and just use that. Any further questions can be directed here, which is VCAA's official communication to students and teachers on what is and isn't okay. One thing to potentially be aware about that's written there - you can glue and staple and tape things into your book if you want, but you must do so in a way that the page doesn't fold outside the book, as fold outs are not allowed
In my BR, I have an a4 page partially pasted into the book at the bottom such that you can pull back the page to view writing on the other side*. Is this allowed?


*The bottom of the page is attached to the BR. The top of the page is loose, but remains within the BR. You can pull back the page from the top that overhangs out of the BR.

keltingmeith

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Re: VCE Methods Question Thread!
« Reply #18731 on: August 25, 2020, 09:53:43 am »
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Not a criticism of your reply - but it would be nice to get some clarification from VCAA about this issue. The question does not state that the volumes in the bottle are restricted to whole numbers of mL, nor does it state that the final rounded answer should be such that less than 2% of bottles are below the allowed volume. Getting an answer of 1003.3 and rounding to the nearest mL should give 1003.

The other reasonable thing to do would be to accept both answers - which has occurred in the past.

I'm going to play devil's advocate and put a vote against both of you. I think the question supplied enough information for the student to think that they should be picking 1004. The students knew from the material supplied that the no more than 2% of bottles should be 2mL under volume - that tells them they HAVE to be rounding up. It doesn't even require particularly difficult thinking - you know that more is better if you have to make a rounding, so you know that you should be rounding up. If a student does not identify this, then their question allows for some proportion of bottles greater than 2% to be underfilled - this implies the student is not thinking of the context of the question when answering it.

As for whether you round to 1003.1, or 1003.12, or whatever is left in this decimal expansion - the question clearly states "the nearest millilitre" - not the nearest tenth of a millilitre, or the nearest hundredth, but the nearest millilitre - so we know we should be rounding to 1003 or 1004. The reason they state the nearest millilitre if we HAVE to round up anyway? Well, I could round up to 1005, and my answer would still be correct for the context supplied - I would still have less than 2% of my bottles underfilled. However, this would no longer be the "nearest" millilitre, hence why the wording is still important. Maybe in the heat of the moment it could seem ambiguous, but I think in a situation with cooler heads, it's very easy to see why 1003 should be interpreted as a wrong answer.

Is it a mean question? For sure, but it's questions like these that differentiate the top students. Do I agree with needing questions to differentiate the top students? No, and I think ranked based models for evaluation are shit and we should remove them for many number of reasons. But thems the games that we are playing.

In my BR, I have an a4 page partially pasted into the book at the bottom such that you can pull back the page to view writing on the other side*. Is this allowed?


*The bottom of the page is attached to the BR. The top of the page is loose, but remains within the BR. You can pull back the page from the top that overhangs out of the BR.

If it can fold outside the margins of the page, then no - it's not allowed. My methods teacher used to talk about the shake test, and I think it definitely captures the spirit of the rule: if you can hold your book by the corner of its front cover and dangle it around, nothing should fall or fold out. If you're still unsure, you should probably post pictures - they're worth a thousand words in situations like these.

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Re: VCE Methods Question Thread!
« Reply #18732 on: August 25, 2020, 11:23:30 am »
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I realise this is perpetuating an OT discussion, but I couldn't resist...

<snip>

KM this makes my head hurt. I appreciate your reasons and that you may just be playing devil's advocate, but it just seems needlessly complicated and tricky. The exact value is some irrational number, whose decimal expansion begins 1003.3... Rounding that number to the nearest integer is 1003. Why shouldn't it be that simple? I don't buy the argument from cohort discrimination. Yes, VCAA need to rank students. No, they don't need to do it in dodgy ways.

Requiring 1004 as the only reasonable answer is suggesting that students interpret the question along the lines of: "The company asks you which smallest whole number of mL they should calibrate their machine to, such that less than 2% of bottles have below the allowed volume". That's a reasonable question, but not what is written. And given VCAA's sanctimony about students being marked on what they write, not what they mean, I think it's reasonable to expect students to answer questions as they are written, not on what we think the question is trying (but failing) to ask.

I also note that in the Specialist 2018 Exam 2, a similar issue arose in Question 5e. The question required finding the smallest mean such that no more than 5% of the distribution was below a specified value. The exact value was something like 146.513... Both 146.51 and 146.52 were accepted.

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Re: VCE Methods Question Thread!
« Reply #18733 on: August 25, 2020, 01:07:04 pm »
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I realise this is perpetuating an OT discussion, but I couldn't resist...

KM this makes my head hurt. I appreciate your reasons and that you may just be playing devil's advocate, but it just seems needlessly complicated and tricky. The exact value is some irrational number, whose decimal expansion begins 1003.3... Rounding that number to the nearest integer is 1003. Why shouldn't it be that simple? I don't buy the argument from cohort discrimination. Yes, VCAA need to rank students. No, they don't need to do it in dodgy ways.

Requiring 1004 as the only reasonable answer is suggesting that students interpret the question along the lines of: "The company asks you which smallest whole number of mL they should calibrate their machine to, such that less than 2% of bottles have below the allowed volume". That's a reasonable question, but not what is written. And given VCAA's sanctimony about students being marked on what they write, not what they mean, I think it's reasonable to expect students to answer questions as they are written, not on what we think the question is trying (but failing) to ask.

I also note that in the Specialist 2018 Exam 2, a similar issue arose in Question 5e. The question required finding the smallest mean such that no more than 5% of the distribution was below a specified value. The exact value was something like 146.513... Both 146.51 and 146.52 were accepted.

I mean, if we're going to argue this purely on what VCAA allow, there's no point continuing the discussion - because the question we're talking about isn't a VCAA question, it's a textbook question, lol. Unless it's popped up on a VCAA paper somewhere and I missed that. But also, I still maintain that rounding up should be a natural assumption based on what you've read from the question - a range is implied, and that range is less than 2%. In fact, the question never asks you to pick a mean so that /exactly/ 2% of bottles will be underfilled, it just asks for a mean so that less than 2% of bottles will be underfilled. The more I read the question, the more I'm convinced that 1005 is a much more suitable answer than 1003 - and previous times this has been asked on the forums seems to agree with me that not picking 1003 should be a natural assumption.

But also yeah if we continue this mods may be forced to split this into a new topic - so I'm happy to continue the discussion, but yeah maybe we should spawn a new topic if we do

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Re: VCE Methods Question Thread!
« Reply #18734 on: August 26, 2020, 08:44:34 am »
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Hey guys ,
I’m struggling to figure out where the 3 comes from in this solution and also where the other 5x went in this solution