S1 MEI - Struggling with hypothesis testing

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ladolcevita8
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#1
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#1
Hey

I'm struggling with some assignment questions on hypothesis testing at S1 level (or S2 for non-MEI ppl). Not sure why but it's hard to wrap my head around what exactly I should do with these types of questions.

I've attached two questions below that are a bit of a struggle (both Q3 + Q4). Could someone please explain what to do, and what in layman's terms is the null and what is the alternative hypothesis?

It'd be fab if someone could explain exactly how you do it step-by-step.

Cheers!
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Kevin De Bruyne
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#2
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(Original post by ladolcevita8)
Hey

I'm struggling with some assignment questions on hypothesis testing at S1 level (or S2 for non-MEI ppl). Not sure why but it's hard to wrap my head around what exactly I should do with these types of questions.

I've attached two questions below that are a bit of a struggle (both Q3 + Q4). Could someone please explain what to do, and what in layman's terms is the null and what is the alternative hypothesis?

It'd be fab if someone could explain exactly how you do it step-by-step.

Cheers!
(Original post by ladolcevita8)
Hey I'm struggling with some assignment questions on hypothesis testing at S1 level (or S2 for non-MEI ppl). Not sure why but it's hard to wrap my head around what exactly I should do with these types of questions. I've attached two questions below that are a bit of a struggle (both Q3 + Q4). Could someone please explain what to do, and what in layman's terms is the null and what is the alternative hypothesis? It'd be fab if someone could explain exactly how you do it step-by-step. Cheers!
This example is classic MEI - unhelpful - as the question is very confusing.

In layman's terms, the null hypothesis is what you assume to be true. For example, when testing for a biased coin you assume that p(heads) = 0.5 and the alternative hypothesis depends on what you're proposing in this example, so if you said that the coin is biased towards heads then you might have alternative as p(heads)>0.5. The alternative hypothesis is something you usually infer from the question.

Here the challenge is identifying the hypotheses - when the company says that they can't distinguish the two brands, you have to infer that p(drinker is correct in identifying what it is) = 0.5, I.e. That they simply guess between two choices at random.

It's confusing because it mentions giving the natural tea and only testing the artificial tea group of 10 people, but as above, what you get is that if people truly can't distinguish between fake and real for the artificial tea, then p(says it's artificial) = 0.5 and p(says it's real) = 0.5, so again in layman's terms you'd expect 5 out of the 10 people to say it's artificial and 5 out of 10 to say it's real, instead you end up with 8/2 split and you have to work out the probability of 8 people saying it's fake and 2 real, assuming the null hypothesis (the 0.5s) is true.

Let me know if you have specific questions or need me to reword anything I've said, otherwise hopefully this clears it up a bit for you.
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ladolcevita8
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#3
(Original post by Kevin De Bruyne)
This example is classic MEI - unhelpful - as the question is very confusing.

In layman's terms, the null hypothesis is what you assume to be true....
Thanks a million for the reply. Some of the MEI questions are a nightmare tbh.

So I've alllllmost got there, I wrote the following:

null hyp. = P = 0.5
alt. hyp. = P > 0.5

X~B(10, 0.5), and we want to find: P( X greater than or equal to 2) since 2 out of 10 couldn't tell it was synthetic.

I checked on the table for n=10, p= 0.5, x=2, and got 0.0547, which is greater than the sig. test of 0.05. (This bits correct)

However, the answer at the back says "we accept the null", but to me the null hypothesis is that it's 50/50.... which means we'd reject the null surely? Where have I gone wrong?

Thanks!
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Kevin De Bruyne
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#4
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#4
(Original post by ladolcevita8)
Thanks a million for the reply. Some of the MEI questions are a nightmare tbh.

So I've alllllmost got there, I wrote the following:

null hyp. = P = 0.5
alt. hyp. = P > 0.5

X~B(10, 0.5), and we want to find: P( X greater than or equal to 2) since 2 out of 10 couldn't tell it was synthetic.

I checked on the table for n=10, p= 0.5, x=2, and got 0.0547, which is greater than the sig. test of 0.05. (This bits correct)

However, the answer at the back says "we accept the null", but to me the null hypothesis is that it's 50/50.... which means we'd reject the null surely? Where have I gone wrong?

Thanks!
Fantastic, like you said you are almost there.

Again in layman's terms, to answer your question and give context around the meaning of a significance level (let's use the 5%)...

Essentially we're trying to prove that a null hypothesis can't be true if something very unlikely happens, assuming the null is true, as a result of a test.

Let's say that I throw a coin who I think is unbiased (0.5 for each head and tail) and I threw 100 as a test and got back 1 heads... safe to say the probability of this happening is very very low (less than 0.05! - key) but that was our null hypothesis, and we have shown that under the null hypothesis it looks very unlikely.

If we got 80 heads and 20 tails instead, our gut would tell us that p(heads) is around 0.8 but anyway we have to test it as it's not so clear cut, and again we find the probability of getting 80 heads out of 100 given we think the probability of gettingbheads is 0.05... now let's say this probability was 0.02, then we say that it's really really low, in fact far too low for the probability to be actually 0.5 for getting a heads.

But what is the boundary for 'really low'? And that is determined by what significance level we use, thankfully it's given in the question most of the time (for Edexcel at least)

So back to your question, you ended up with 0.0547, and our boundary for rejecting the null is when the probability is *less* than or equal to 0.05, which is what I was illustrating with the 100 coins example.

So whilst 0.0547 is low, it's not 'unlikely' enough for us to say that it's looking like the null is wrong. If we found it was 0.045 then it would be too unlikely when we compare it to 0.05 (5%)

So when you calculate the probability under the hull hypothesis, reject it if it's less than the significance level
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ladolcevita8
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#5
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(Original post by Kevin De Bruyne)
Fantastic, like you said you are almost there.

Again in layman's terms, to answer your question and give context around the meaning of a significance level (let's use the 5%)...

Essentially we're trying to prove that a null hypothesis can't be true if something very unlikely happens, assuming the null is true, as a result of a test.

Let's say that I throw a coin who I think is unbiased (0.5 for each head and tail) and I threw 100 as a test and got back 1 heads... safe to say the probability of this happening is very very low (less than 0.05! - key) but that was our null hypothesis, and we have shown that under the null hypothesis it looks very unlikely.

If we got 80 heads and 20 tails instead, our gut would tell us that p(heads) is around 0.8 but anyway we have to test it as it's not so clear cut, and again we find the probability of getting 80 heads out of 100 given we think the probability of gettingbheads is 0.05... now let's say this probability was 0.02, then we say that it's really really low, in fact far too low for the probability to be actually 0.5 for getting a heads.

But what is the boundary for 'really low'? And that is determined by what significance level we use, thankfully it's given in the question most of the time (for Edexcel at least)

So back to your question, you ended up with 0.0547, and our boundary for rejecting the null is when the probability is *less* than or equal to 0.05, which is what I was illustrating with the 100 coins example.

So whilst 0.0547 is low, it's not 'unlikely' enough for us to say that it's looking like the null is wrong. If we found it was 0.045 then it would be too unlikely when we compare it to 0.05 (5%)

So when you calculate the probability under the hull hypothesis, reject it if it's less than the significance level

Ahhh of course, it makes perfect sense now. .

Thanks a lot for your help again!

If you've got the time would you mind having a look at Q4 for me. I'm trying to set it up but I'm unclear on what the null and alt. should actually be.

I want to say that the null = P > 0.5 (since we're assuming that the saying is true - buttered toast is more likely to land butter-side up and there's only two ways it usually lands) and the alternative would be 11/18?

But, this feels wrong 'cause 11/18 is actually over 0.5 so isn't an alternative, haha. Sorry about this!
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Kevin De Bruyne
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#6
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(Original post by ladolcevita8)
Ahhh of course, it makes perfect sense now. .

Thanks a lot for your help again!

If you've got the time would you mind having a look at Q4 for me. I'm trying to set it up but I'm unclear on what the null and alt. should actually be.

I want to say that the null = P > 0.5 (since we're assuming that the saying is true - buttered toast is more likely to land butter-side up and there's only two ways it usually lands) and the alternative would be 11/18?

But, this feels wrong 'cause 11/18 is actually over 0.5 so isn't an alternative, haha. Sorry about this!
No worries, I see what you're doing there but:

The alternative hypothesis isn't based on the test statistic in that way (as in, they're set before running the test so they won't have the alternative as 11/18)

The null hypothesis is usually a '... = ....' and the alternative is an inequality. This makes sense because it'd be difficult, if the null hypothesis were p>0.5 and alternative were p<0.5, it would be impossible to run a test on this in a meaningful way (from an S1 perspective at least) so just like in Q3, we need to use an = in our null and interpret what the imequalty for the alternate hypothesis should be (in this case it's that the buttered side is more likely to land face up or face down, whichever way round it was) so we use H0: p = 0.5 and H1: p>0.5.

Key takeaways from this are that The inequality from H1 is not based on the test statistic, and H0 (null hypothesis) is always a ' ... = ....'

It's always good to help someone who is thinking for themselves and asking the right questions
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ladolcevita8
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#7
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(Original post by Kevin De Bruyne)
No worries, I see what you're doing there but:

.....
(I'm trying to think for myself !)

Ok so rule of thumb, null is always "equals", and alternative is always "less than, equal to, or greater than". And the null/alt. are set before any tests are ran (E.g. these were set before the 18 students decided to test the claim).

Thus: null = 0.5, alternative > 0.5. => X ~ B (18, 0.5)

But I'm unsure where to go from here. I thought I had it but it was the wrong answer. I can see that we will check the table for n=10, p=0.5 and x = 7 in order to get 0.240 .... but I don't know why we checked 7 (is it to do with 11/18? 7/18 = q?)

(Sorry about this again - being a bit of a pain)
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Kevin De Bruyne
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(Original post by ladolcevita8)
(I'm trying to think for myself !)

Ok so rule of thumb, null is always "equals", and alternative is always "less than, equal to, or greater than". And the null/alt. are set before any tests are ran (E.g. these were set before the 18 students decided to test the claim).

Thus: null = 0.5, alternative > 0.5. => X ~ B (18, 0.5)

But I'm unsure where to go from here. I thought I had it but it was the wrong answer. I can see that we will check the table for n=10, p=0.5 and x = 7 in order to get 0.240 .... but I don't know why we checked 7 (is it to do with 11/18? 7/18 = q?)

(Sorry about this again - being a bit of a pain)
Oh, I meant the thinking for yourself bit as a compliment rather than an insult as it is what you are doing.

Rules of thumb look good.

To answer your question, I haven't attempted it myself but it looks like it's to do with reading from the tables - as you identified, q!

You can either have p or q let's call them, where p is the 11 way of looking at things and q is the 6 7 way of looking at things. But again, let's take the simple example of a coin.

If p is the event that we flip a heads and q a tails, we can look at a test in one of two ways:

Let's say we flip a coin 100 times, and we're looking to understand if the coin is biased towards heads or not, that is: H0: p(heads) = 0.5 vs H1: p(heads) > 0.5.

And let's say we end up with 80 heads, I.e. p = 80 and q = 100-80=20.

To work out the probability of getting this result or a more extreme value (I.e. More than 80 heads in this case) we can either look at p(getting more than 80 heads), I already foresaw confusing p's but anyway: p(p>80)), or we can look at it from the tails point of view, p(q<20).

This is done because these two are the same event.

For example, if you get 81 heads it's the same as saying you get 19 tails, if I give you one piece of information plus the total you can deduce the other because it's binary (one or the other) so naturally the probabilities of these two things are equal.

The same is going on in your example, if you get the same answer in the end I think either way would be acceptable, but I think it's done because one way the tables are easier to read than the other.
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ladolcevita8
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#9
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(Original post by Kevin De Bruyne)
Oh, I meant the thinking for yourself bit as a compliment rather than an insult as it is what you are doing....
Haha oh I know you meant it that way! I worded it poorly so it sounded negative - never mind

Anywho, to summarise the coffee question:

null: P = 0.5 (since like you said, its a random guess of yes or no)
alt.: P < 0.5 (since critics assume its actually lower)

X ~ B(10, 0.5)

P (X less than or equal to 2) = 1 - P (X is more than or equal to 3)

look on distribution table where n = 10, x = 7, p = 0.5 - we get 0.9453

Thus 1 - 0.9453 = 0.0547

0.0547 > 0.05, thus we accept null hyp.



You're an absolute star - I had a "ping" moment and my mind is actually absorbing it now so thanks a million!
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