# central limit therom help needed

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#1
https://i.imgur.com/jUvEXm3.jpg
so for part b I said we assumed that central limit theorem is used since sample is large. but the markschme also adds no need to assume normal distribution. doesnt the central limit therom also assume normal distribution hence doesn't this statement contradict it self?
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#2
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2 years ago
#3
(Original post by assassinbunny123)
ghostwalker
vc94
Gregorius
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No, the "best" example is the bean machine
https://en.wikipedia.org/wiki/Bean_machine
where at each level, the ball bounces left or right according to a binomial distribution, but the distribution at the bottom is approximately normal.
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2 years ago
#4
(Original post by assassinbunny123)
doesnt the central limit therom also assume normal distribution?
No, and this is an important point, as it explains why you get to use the normal distribution so much in probability and statistics!

There are various forms of the central limit theorem, but a straightforward one states that if you draw samples of size n from any probability distribution with finite mean and variance, then the means of those samples will be normally distributed.

If you want an example where the central limit theorem does not hold, try the Cauchy Distribution - a distribution where the mean and variance do not exist!
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2 years ago
#5
(Original post by assassinbunny123)
https://i.imgur.com/jUvEXm3.jpg
so for part b I said we assumed that central limit theorem is used since sample is large. but the markschme also adds no need to assume normal distribution. doesnt the central limit therom also assume normal distribution hence doesn't this statement contradict it self?
Have a look at galton boards, its a neat way of demonstrating the central limit theorem.

I recommend watching a video from the youtube channel DONG, its quite cool.

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2 years ago
#6
(Original post by mqb2766)
No, the "best" example is the bean machine
https://en.wikipedia.org/wiki/Bean_machine
where at each level, the ball bounces left or right according to a binomial distribution, but the distribution at the bottom is approximately normal.
No, not quite.

At each level, the ball bounces left or right according to a Bernoulli distribution, and since the sum of n independent Bernoulli random variables is a binomial distribution, the distribution at the bottom is Binomial - which can then (after a fair amount of algebra!) be shown to converge (in a sense that needs to be made rigourous) to a normal distribution.

So these little machines demonstrate the special case of the central limit theorem for Bernoulli random variables.
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#7
(Original post by Shaanv)
Have a look at galton boards, its a neat way of demonstrating the central limit theorem.

I recommend watching a video from the youtube channel DONG, its quite cool.

pretty cool tho, but it didn't answer my question
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2 years ago
#8
so whatever kind of distribution the parent population has, you will always find that the distribution of sample means is normal.
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2 years ago
#9
My stats lecturer basically said if the number of trials is greater than or equal to 30, the distribution (whatever it was originally, binomial, bernoulli etc.) can be approximated by a normal distribution now.
I'm careful with the wording as perhaps that is different to "assuming" the variable is normally distributed. I don't think weight loss is "naturally" normally distributed, but we can approximate it as, with a large enough sample size
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#10
(Original post by Gregorius)
No, and this is an important point, as it explains why you get to use the normal distribution so much in probability and statistics!

There are various forms of the central limit theorem, but a straightforward one states that if you draw samples of size n from any probability distribution with finite mean and variance, then the means of those samples will be normally distributed.

If you want an example where the central limit theorem does not hold, try the Cauchy Distribution - a distribution where the mean and variance do not exist!
but you just said that central limit therom says that the distribution of mean and variance are normally distributed. here the answer says that central limit therom can be assumed but normal distribution cannot,how is this possible?
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2 years ago
#11
(Original post by assassinbunny123)
but you just said that central limit therom says that the distribution of mean and variance are normally distributed. here the answer says that central limit therom can be assumed but normal distribution cannot,how is this possible?
Gregorius is trying to help you. the population distribution does not have to be normal; the distribution of sample means is normal.
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#12
(Original post by the bear)
Gregorius is trying to help you. the population distribution does not have to be normal; the distribution of sample means is normal.
sorry Gregorius if I didn't understand what you meant before. I understand now thank you.
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2 years ago
#13
(Original post by assassinbunny123)
but you just said that central limit therom says that the distribution of mean and variance are normally distributed. here the answer says that central limit therom can be assumed but normal distribution cannot,how is this possible?
I'm getting a bit confused about what you're confused about, so let's start from scratch!

Suppose that you start with a probability distribution that has a finite mean (mu) and variance (sigma-squared). If you draw repeated samples of size n from that distribution, then the distribution of the means of those samples will be approximately normal with mean mu and variance sigma-squared divided by n. That's the central limit theorem.

Let's turn to the question you posted. Here you have a single random sample of size 80 from one underlying population and a single sample of size 65 from another underlying population and you're asked whether these two samples give sufficient evidence to show that the means of the underlying populations are different.

To do this, you use the means of each sample - the central limit theorem implies that these can be taken as having come from normal distributions.

So, you don't need to assume normality of the distributions of weight losses, as the central limit theorem tells you that the means of these distributions will be (approximately) normally distributed.
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