S3 question - assumptions

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Hi guys ,

I’m stuck on the questions I have circled . I’m trying to be careful in the wording for these assumption questions as they are quite specific . What I don’t understand is when we do and don’t assume sample means are normally distributed. Like in 19b , the answer was about how we don’t assume weight loss follows a normal distribution . Why don’t we assume this ?

Similar sort of thing for the next question . The answer was that sample sizes are large so CLT guarantees sample means are approx normally distributed. Why is this guaranteed and when wouldn’t it be guaranteed? It also says no assumptions are being made

Thanks a lot :smile:

Reply 1

Prasiortle

13

Original post by Angels1234

https://postimg.cc/gallery/umtprcvg/

Hi guys ,

I’m stuck on the questions I have circled . I’m trying to be careful in the wording for these assumption questions as they are quite specific . What I don’t understand is when we do and don’t assume sample means are normally distributed. Like in 19b , the answer was about how we don’t assume weight loss follows a normal distribution . Why don’t we assume this ?

Similar sort of thing for the next question . The answer was that sample sizes are large so CLT guarantees sample means are approx normally distributed. Why is this guaranteed and when wouldn’t it be guaranteed? It also says no assumptions are being made

Thanks a lot :smile:

I'm afraid I don't know enough statistics to help with anything beyond S2, so I'll tag some people who will be more conversant in this area:

@Gregorius
@Notnek

Reply 2

Gregorius

14

Original post by Angels1234

https://postimg.cc/gallery/umtprcvg/

Hi guys ,

I’m stuck on the questions I have circled . I’m trying to be careful in the wording for these assumption questions as they are quite specific . What I don’t understand is when we do and don’t assume sample means are normally distributed.

As far as you're concerned, as long as sample sizes are large enough, sample means will be (approximately) normally distributed.

The central limit theorem tells you that if you draw samples of size n from an underlying population, then the mean of those samples will be distributed closer and closer to a normal distribution as n get larger and larger. This applies whatever the original distribution of observations was (provided that the original distribution meets some mathematical requirements).

Like in 19b , the answer was about how we don’t assume weight loss follows a normal distribution . Why don’t we assume this ?

Samples sizes are large; therefore apply the central limit theorem.

Similar sort of thing for the next question . The answer was that sample sizes are large so CLT guarantees sample means are approx normally distributed. Why is this guaranteed and when wouldn’t it be guaranteed? It also says no assumptions are being made

This is stuff you don't need to know at your level, but there are various forms of the central limit theorem that demand that various conditions should apply to the underlying distribution for the result to follow. One example of these conditions is for the underlying distribution to have a finite mean and a finite variance.

Reply 3

Angels1234

OP

14

Original post by Gregorius

This is stuff you don't need to know at your level, but there are various forms of the central limit theorem that demand that various conditions should apply to the underlying distribution for the result to follow. One example of these conditions is for the underlying distribution to have a finite mean and a finite variance.

The thing is usually in questions where the sample sizes of the 2 populations are large it asks the relevance of the central limit theorem . I normally go for like as n is sufficiently large CLT allows us to assume the sample means are normally distributed. So how would our answer change if N was smaller . Because we only assume sample means are approx normally distributed providing n is large .
.. and this is where my confusion comes in . Is N is sufficiently large how do we know when to say hat sample means are normally distributed and when to say we don’t assume sample means are normally distributed. Does it no longer become an assumption after n gets very big .

Thanks again :smile:

Reply 4

Gregorius

14

Original post by Angels1234

The thing is usually in questions where the sample sizes of the 2 populations are large it asks the relevance of the central limit theorem . I normally go for like as n is sufficiently large CLT allows us to assume the sample means are normally distributed. So how would our answer change if N was smaller . Because we only assume sample means are approx normally distributed providing n is large .

If the sample size is small, then you can't necessarily apply the central limit theorem. At the level you're answering questions you won't have to deal with this, other than to point out possible problems if n is small.

The modern approach to small sample size problems is to use computer simulation to calculate a numerical approximation to the sampling distribution to the mean. That is, actually draw many small samples from a population, calculate their means and use their empirical distribution as the sampling distribution of the mean.

.. and this is where my confusion comes in . Is N is sufficiently large how do we know when to say hat sample means are normally distributed and when to say we don’t assume sample means are normally distributed. Does it no longer become an assumption after n gets very big .

Yes basically, if n is big enough for the CLT to work, you don't need to assume its result!

S3 question - assumptions

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