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# S2 OCR, Binomial Hypothesis Testing Question

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1. Hi, how do I go about solving this?
I've seen the mark scheme but don't get it so I will need explanation at all steps.

So far this is what I've thought:

Ho: p=0.65
H1: p<0.65

X~B(2n,0.65)

Since we want to reject,
P(X>=n)<0.15
so
P(X<=n-1)>0.85

and I can't use tables to solve that?
2. (Original post by DQd)
Ho: p=0.65
H1: p<0.65
Absolutely, well done.

X~B(2n,0.65)
On the right track.

Since we want to reject,
P(X>=n)<0.15
so
P(X<=n-1)>0.85

and I can't use tables to solve that?
That's where you went wrong. Since this is a lower-tailed test or whatever it's called you need P(X <= n) > 0.15 -- then you need to do it by inspection. Remember you want the smallest n such that the test is rejected.
3. (Original post by Zacken)
That's where you went wrong. Since this is a lower-tailed test or whatever it's called you need P(X <= n) > 0.15 -- then you need to do it by inspection. Remember you want the smallest n such that the test is rejected.
Can I ask why we use that and why what I posted doesn't work?

"
P(X>=n)<0.15
so
P(X<=n-1)>0.85
"

Doesn't this mean that the probability of 'x is = to or more than the actual number (n) that agreed to the shopping mall' is less than 15%, and in that case it would be rejected and that's what we need?

Also I don't get how to use the tables to satisfy
P(X <= n) > 0.15
and
x~B(2n, 0.65)

Mark scheme says x~b(18,0.65) (n=9) but doesn't x~b(6,0.65) (n=3) also work?
P(x<=3)=0.3529 which >0.15??
4. (Original post by DQd)
Can I ask why we use that and why what I posted doesn't work?

Think back to your definition of a hypothesis test, it's a lower-tailed one. For it to be rejected, we need substantial evidence that the proportion of people is less than 65%.

Anywho, for the hypothesis to fail, you need to have P(X <= n) < 15% because that's saying that if (assuming that X ~ B(2n, 0.65), i.e: assuming that the proportion is 65%) then if the probability that there are less than n people agreeing is smaller than the significance level and hence the null hypothesis is rejected. You see what I'm getting at? Here's a video about hypothesis testing with the binomial distribution that you might find helpful for understanding.

Think about it, X = 0 is the worst case, nobody agrees even though the claim says that 65% agrees. X=1 is also bad, all the way up to X=n is bad.

BTW, typo in my above post - should be P(X <= n) < 15% which is why n=9 is the smallest value. Sorry.
5. (Original post by Zacken)

Think back to your definition of a hypothesis test, it's a lower-tailed one. For it to be rejected, we need substantial evidence that the proportion of people is less than 65%.

Anywho, for the hypothesis to fail, you need to have P(X <= n) < 15% because that's saying that if (assuming that X ~ B(2n, 0.65), i.e: assuming that the proportion is 65%) then if the probability that there are less than n people agreeing is smaller than the significance level and hence the null hypothesis is rejected. You see what I'm getting at? Here's a video about hypothesis testing with the binomial distribution that you might find helpful for understanding.

Think about it, X = 0 is the worst case, nobody agrees even though the claim says that 65% agrees. X=1 is also bad, all the way up to X=n is bad.

BTW, typo in my above post - should be P(X <= n) < 15% which is why n=9 is the smallest value. Sorry.
It all works when I use 'P(X <= n) < 15%' .

It makes more sense to be less than when I think about the reject region rather than use words, since if it is a smaller probability than 15% it will fit in the region.

Still can't get my head around:
"if the probability that there are less than n people agreeing is smaller than the significance level and hence the null hypothesis is rejected. "

Because I'm thinking if the chance less than n people agree is small, the chance of 'at least 65%' is true goes up.

The mark scheme is pretty useless, but is it wrong because it uses more than?
6. (Original post by DQd)
It all works when I use 'P(X <= n) < 15%' .

It makes more sense to be less than when I think about the reject region rather than use words, since if it is a smaller probability than 15% it will fit in the region.

Still can't get my head around:
"if the probability that there are less than n people agreeing is smaller than the significance level and hence the null hypothesis is rejected. "

Because I'm thinking if the chance less than n people agree is small, the chance of 'at least 65%' is true goes up.

The mark scheme is pretty useless, but is it wrong because it uses more than?
Nah, the markscheme seems to be using the "accept" region, notice how it picks the first p = 0.13597 that's <= 15%? I'll get back to your other questions in a bit, gotta go now

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